{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Characterization of Systems in the Spectral Domain\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Transfer Function\n",
    "\n",
    "Different time-domain representations are used to characterize linear time-invariant (LTI) systems. For instance, the constant coefficients of ordinary partial differential equations (ODEs), the [impulse response](../systems_time_domain/impulse_response.ipynb) and the [step response](../systems_time_domain/step_response.ipynb). The [transfer function](../systems_time_domain/eigenfunctions.ipynb#Transfer-Function) denotes the complex factor $H(s)$ an eigenfunction $e^{s t}$ at the input of an LTI system is weighted with when passing through the system. Since the Laplace transform decomposes signals with respect to these eigenfunctions, the transfer function constitutes a representation of an LTI system in the spectral (Laplace or Fourier) domain. The links between the transfer function and the time-domain representations of LTI systems are discussed in this section."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Relation to Impulse Response and In-/Output Signal\n",
    "\n",
    "It was already outlined in the context of the [convolution theorem of the Fourier](../fourier_transform/theorems.ipynb#Convolution-Theorem) and [Laplace transform](../laplace_transform/theorems.ipynb#Convolution-Theorem), that an LTI system can be represented equivalently in the temporal and spectral domain. In the temporal domain the output $y(t)$ of an LTI system is given by convolving the input signal $x(t)$ with the [impulse response](../systems_time_domain/impulse_response.ipynb) $h(t)$ of the system\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = x(t) * h(t)\n",
    "\\end{equation}\n",
    "\n",
    "Applying the convolution theorem of the Laplace transform yields\n",
    "\n",
    "\\begin{equation}\n",
    "Y(s) = X(s) \\cdot H(s)\n",
    "\\end{equation}\n",
    "\n",
    "where $X(s) = \\mathcal{L} \\{ x(t) \\}$ and $Y(s) = \\mathcal{L} \\{ y(t) \\}$ are the Laplace transforms of the input and output signal, respectively. The transfer function $H(s)$ is given as the Laplace transform of the impulse response $h(t)$\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) = \\mathcal{L} \\{ h(t) \\}\n",
    "\\end{equation}\n",
    "\n",
    "This may also be concluded from the [previously derived link between the impulse response and the transfer function](../systems_time_domain/eigenfunctions.ipynb#Link-between-Transfer-Function-and-Impulse-Response). The derived results show that an LTI system can be fully characterized either in the temporal or spectral domain by its impulse response $h(t)$ or transfer function $H(s)$ or $H(j \\omega)$, respectively.\n",
    "\n",
    "![LTI system in the temporal and spectral domain](LTI_system_time_spectral_domain.png)\n",
    "\n",
    "It can furthermore be concluded that the transfer function is given as the quotient between output $Y(s)$ and input signal $X(s)$ in the Laplace domain\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) = \\frac{Y(s)}{X(s)}\n",
    "\\end{equation}\n",
    "\n",
    "for $X(s) \\neq 0$. The same relations hold in the Fourier domain if the region of convergence (ROC) of the impulse response includes the imaginary axis $\\Re \\{ s \\} = 0$. In general, the transfer function can be derived by divison of the spectra of the output and input signal. This can be used to measure the transfer function of a system by specific input signals that fulfill $X(s) \\neq 0$ with $X(s) \\approx$ constant."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The [transfer function $H(s)$ of the 2nd order low-pass filter](../systems_time_domain/network_analysis.ipynb#Transfer-Function) was derived as\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) = \\frac{1}{C L s^2 + C R s + 1}\n",
    "\\end{equation}\n",
    "\n",
    "According to above findings, the impulse response $h(t)$ of the low-pass is given by inverse Laplace transform of the transfer function $H(s)$. First the transfer function is defined in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{C L s^{2} + C R s + 1}$"
      ],
      "text/plain": [
       "        1         \n",
       "──────────────────\n",
       "     2            \n",
       "C⋅L⋅s  + C⋅R⋅s + 1"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "%matplotlib inline\n",
    "sym.init_printing()\n",
    "\n",
    "R, L, C = sym.symbols('R L C', positive=True)\n",
    "s = sym.symbols('s', complex=True)\n",
    "\n",
    "H = 1/(C*L*s**2 + C*R*s + 1)\n",
    "H"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then the inverse Laplace transform is computed for the specific normalized values $L = .5$, $R = 1$, $C = .4$ of the network elements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{5 e^{- t} \\sin{\\left(2 t \\right)} \\theta\\left(t\\right)}{2}$"
      ],
      "text/plain": [
       "   -t                      \n",
       "5⋅ℯ  ⋅sin(2⋅t)⋅Heaviside(t)\n",
       "───────────────────────────\n",
       "             2             "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t', real=True)\n",
    "RLC = {R: 1, L: sym.Rational('.5'), C: sym.Rational('.4')}\n",
    "\n",
    "h = sym.inverse_laplace_transform(H.subs(RLC), s, t)\n",
    "h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This result is plotted for illustration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": "JVBERi0xLjQKJazcIKu6CjEgMCBvYmoKPDwgL1BhZ2VzIDIgMCBSIC9UeXBlIC9DYXRhbG9nID4+CmVuZG9iago4IDAgb2JqCjw8IC9FeHRHU3RhdGUgNCAwIFIgL0ZvbnQgMyAwIFIgL1BhdHRlcm4gNSAwIFIKL1Byb2NTZXQgWyAvUERGIC9UZXh0IC9JbWFnZUIgL0ltYWdlQyAvSW1hZ2VJIF0gL1NoYWRpbmcgNiAwIFIKL1hPYmplY3QgNyAwIFIgPj4KZW5kb2JqCjEwIDAgb2JqCjw8IC9Bbm5vdHMgWyBdIC9Db250ZW50cyA5IDAgUgovR3JvdXAgPDwgL0NTIC9EZXZpY2VSR0IgL1MgL1RyYW5zcGFyZW5jeSAvVHlwZSAvR3JvdXAgPj4KL01lZGlhQm94IFsgMCAwIDQyMi44MzI1IDI5My4wNTEwNjI1IF0gL1BhcmVudCAyIDAgUiAvUmVzb3VyY2VzIDggMCBSCi9UeXBlIC9QYWdlID4+CmVuZG9iago5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTEgMCBSID4+CnN0cmVhbQp4nIWZy64ctxGG9/0UvZQXongvchnBsYDsZAnJIvbCkGXZgkaBIjt6/XzFbvbMnKnBCDjQOfXXz2KTde0O68clrB9Wv37k59v67/Vn/v91Desrfj4snr9OS47RtRQLf3w6/xF7cr4EXyNSf/3n78vy2/LibyzxFdKrZSnRhZICtOCd6Jq+uyhnyadLSZTqWtCFzryzbF88bot/YKNs3rWL7WNSkQt2zS6HUiVj+kLqXd+kn5blJSfxbXn5dn3xQ2BP69vfltxcl5STlDV312LoMbWW8/r21+WZ/259+3H9+9tl2F5CTK6IPDV2KX5gLUTvUpc75uITc70478ONuQvxI3M9qI075vK1uViry7E8NXcpfmAu1uRiLnfM1WtzKYlruT81dyl+YC6l7Grtd8y1a3MXrndp7lL8yFN8RME2Fg5P+QLRr889Lpu9zLVTxR+9z20w3p1YWQPP9TZ2zxIvfoi7nWd/6loLa73e93725sQBl9BS0a03V59Kj51fbAOXaxKaaDg7bqeX3lvZNnHxeM0laZ3Y1hwwnN/FuZEvbMTrgrrn/Rf4L75///GXf/715pfPX5+f/vj819f1+/+w6dttX534se2z9I47cTW56sa5msoT9HaOTPckNs/GCI8ivYR0ZewsfWSMmJGGK4u0JJcnYRkLsTrCOZV4Ze1C/MhcINn2XLPU2tJuLt81V8R1L8T1tbmz+KG5kl0IPoVSWpz26l17rTvW9bFf2zuLH9prhSDgnnNuoe722j17EWeNKbfaruxdiB/Zi4F0laT3lJrvW3De9xXChnzcxMu1vbP4ob0srnqCL8bm427vcJcrkgbjcw2ekJ10jftYxYnE8fvb0yjFusDvPz3786fvjoyyPqmsidumSmpipgqs/32//mv9vAZXKI8ehJhqNbIkjlT3f8Kjiy+NOy/c2o+vVoL3f3+8e//jq5fru6/3ymhhhV4TTtq1TNYRPW8sOcrNSSRGuXJOvBSs7toGcFo0pr2ITyQo8rj3LeyrmwiE6CRUMibuUJ3nFDZ3eGMjEJLLrUc9qhTHkgfBQiBkF1sTwoLUzpKplrITLARC4dREOl6QXamp+r7rGwDqXBhFIbNKwaEJ15x3fQsZBMqbLxUxYeuDNDkItwgEnLF4fo0FhV7jvAMLQL05n3MYYvJ/z02mvoUMQuolFhV3V9inHPo3wFCXkiRyMVxqyQ3rh/4tAqG7UKV4DQ0cmjvy84QsZBCylNI5OLyArF3zmXCLDIJ031of4lZrOrZkIadFKDU9ND0J3IabJA9tBBMZBG4ml03cIh1RPQi3yCBIbL3rWXCXuft4JtwiEAgjkaQBotGaSvNhJ1jIIKSe6kgaGlFdji1ZyCAIyaKVISbT+362cItAiMSd7zHv4rn8jXioRlLnOGdMCnk6Hdq3yCBUgrVte9TesOWDcItASEREjGXbY2jVtz0uTWQQuArf+34xqRQ5CLfIILRKDm9DTGqoF4RbBELGQSibZTgXO/XTT01kEIQS1MLwd/ITbnAQbhEIpBzucVQVylyJLU4LFjIIQrcdt5DN1KJUD8ItAkErqy+d06YC5aDxvhMsBAJ1Sjwt+khSXfg3Q8dCBqGXEoKmQTyHxHz4qYVA6C611ISd5kJGFPqMnWAhp6WRbEgeQcUk2kgjFDeCiUAgskuTxKNFXKyGw5dMBEIjncVM3tEuqZN19me2gNPSuX7OunS6guYy/XncN2QiEPpIZl3WQB9Rc6htvwUTYV70ibkl5NLWxj2FgHduBBNRQtWMH3td8WacLM8yaCJKaK752rR7CK5HoWZMgoFACIFsJp79cZuphLrfmgWougZGrxJXui165XCsbyFKSLTfoXHCmVvqWrsnwUCUQIwkX1NYE45DjvDzkS1ECZWC2mh8UsHRgpQw9W8BVRf1QY+z0HAwq81exwJUXfvr3EgdyeMF+sZg6hvIabxXoK/L7DKyIDmh7YFjIkogdUiXkTp5rDBTqgWoOkOK3zomzoMmbl6AAag6yVUifkswEa1c/LG8gSihMMr7RC2JOszOembIVVmzRyYJruNFVcFzp7qBKIHQC9q0qVj6cTRPxarKbF95jGHQM0n46TkWAoFrCyRhsos+kBB+81EtRAnZtZ79yN+4Vuz+IBiIEroGXS64VNAjm+XSAlDXdMclM4yQx1Jgjp2nYyEQGHZKRyyrTkC1MGntBAtRAs0bs09o6+gNQ40HwUAg0A6XmJkBVn6j8+4zS5sIBLoMHqf4utKVURvpiHeChShB6JopvLLSE5A1yhFgFgKB7M3IzaigmczXXI+HthB960Yvlnm4oOtoZ9z3PG0iSqjMrEKfsYomNfB5DxZyWqInQ3rOoumjMS0zU24EE1FCgx065Z/2vpR8TFwmAkFzGS1MjXraWkPzJFgIBE0GjC+UFG2XqtTpqyaiBOYMzJJtRvdTSw2TYCAQkk4vjISrDnE1kql2fQNAPeOR9G9hiJGGeQsmogSSGL08+VjXScxAaRIMBAI+T6GlC1NxZ5uzvJqIEpjhk/ZvOgTQ3B+3ZgCn8Y5UO6WqUrJxnaXPAlSdfB89o7ues1dfaVPfQCDQHjbu3jcVp4zG3I+FQGiEBd1bGjfJ1R9OZACqTiagIyxjFcm+zPM0ANS7vhtnxOhD2nCYubyFKIHHSjxc3sR03/P8LeS0JH1HQCeprqt2a5mNsokogbRJyfVDXPvs3A05ytovxE47PISk276fv4kogYBjrKhDzKTEnU6CgUDQShV97OPcmJzrTI4mogQ6HZ16xqsakrLMBzAAfaFOI9Urfcz2akfou3d9C9newBfaU5KS+olwEGESDARCZp/el11ccsrzkS1ECayj1TltnhiOFzMmAoHQIyL2k0hqeT6DhSih0m3qO55tHZy3TIKBQNC3/z1qWVGp3uqmfitXZapWZcaT7bEYXuYJWQgE8nwPtNqbWM5OZACqTmOCu8Th6YSIn1nURCA0/ghb5GUageM4b+WqzJxQaHf2a6dbPdQNBAKR2scL1SHeysRGsBAl0PWTKmPaxMxhc/8Wol9htsuWbR0mr+lBJnJ6+jUn6SuAof96/bLG9R/b186rD4b33ncerd+F6PIVxvGJ89v4zPn0o+bJ/qi5fD4+X45dKPXO9xDrw5Rl+fY76X3br5f/A8yB5rwKZW5kc3RyZWFtCmVuZG9iagoxMSAwIG9iagoyMzczCmVuZG9iagoxNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE3OCA+PgpzdHJlYW0KeJw9kEsSAyEIRPeeoo8gP9HzJJXV5P7bNM6YhfYTKGywpegYzsvVkF3xllYYnvg27wueiaup9YckFTL7pg4Zi2fyFCckFK8mY0DcsdXWo3Eywgy7LCZmYK6Kr0QuqFAC6lHyamaj4Gqmgr59xFaZ9RYXCEUCJixnK/6fA9PrFh0MFpjtN8uzPPYK03QszHuGeQ9FI34G5VpuYGWtg5+GHZr3tmgo/+QnexZ4tc8P3qtCoQplbmRzdHJlYW0KZW5kb2JqCjE0IDAgb2JqCjw8IC9CYXNlRm9udCAvRGVqYVZ1U2Fucy1PYmxpcXVlIC9DaGFyUHJvY3MgMTUgMCBSCi9FbmNvZGluZyA8PCAvRGlmZmVyZW5jZXMgWyAxMTYgL3QgXSAvVHlwZSAvRW5jb2RpbmcgPj4gL0ZpcnN0Q2hhciAwCi9Gb250QkJveCBbIC0xMDE2IC0zNTEgMTY2MCAxMDY4IF0gL0ZvbnREZXNjcmlwdG9yIDEzIDAgUgovRm9udE1hdHJpeCBbIDAuMDAxIDAgMCAwLjAwMSAwIDAgXSAvTGFzdENoYXIgMjU1IC9OYW1lIC9EZWphVnVTYW5zLU9ibGlxdWUKL1N1YnR5cGUgL1R5cGUzIC9UeXBlIC9Gb250IC9XaWR0aHMgMTIgMCBSID4+CmVuZG9iagoxMyAwIG9iago8PCAvQXNjZW50IDkyOSAvQ2FwSGVpZ2h0IDAgL0Rlc2NlbnQgLTIzNiAvRmxhZ3MgOTYKL0ZvbnRCQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRm9udE5hbWUgL0RlamFWdVNhbnMtT2JsaXF1ZQovSXRhbGljQW5nbGUgMCAvTWF4V2lkdGggMTM1MCAvU3RlbVYgMCAvVHlwZSAvRm9udERlc2NyaXB0b3IgL1hIZWlnaHQgMCA+PgplbmRvYmoKMTIgMCBvYmoKWyA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMAo2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDMxOCA0MDEgNDYwIDgzOCA2MzYKOTUwIDc4MCAyNzUgMzkwIDM5MCA1MDAgODM4IDMxOCAzNjEgMzE4IDMzNyA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2CjYzNiA2MzYgMzM3IDMzNyA4MzggODM4IDgzOCA1MzEgMTAwMCA2ODQgNjg2IDY5OCA3NzAgNjMyIDU3NSA3NzUgNzUyIDI5NQoyOTUgNjU2IDU1NyA4NjMgNzQ4IDc4NyA2MDMgNzg3IDY5NSA2MzUgNjExIDczMiA2ODQgOTg5IDY4NSA2MTEgNjg1IDM5MCAzMzcKMzkwIDgzOCA1MDAgNTAwIDYxMyA2MzUgNTUwIDYzNSA2MTUgMzUyIDYzNSA2MzQgMjc4IDI3OCA1NzkgMjc4IDk3NCA2MzQgNjEyCjYzNSA2MzUgNDExIDUyMSAzOTIgNjM0IDU5MiA4MTggNTkyIDU5MiA1MjUgNjM2IDMzNyA2MzYgODM4IDYwMCA2MzYgNjAwIDMxOAozNTIgNTE4IDEwMDAgNTAwIDUwMCA1MDAgMTM1MCA2MzUgNDAwIDEwNzAgNjAwIDY4NSA2MDAgNjAwIDMxOCAzMTggNTE4IDUxOAo1OTAgNTAwIDEwMDAgNTAwIDEwMDAgNTIxIDQwMCAxMDI4IDYwMCA1MjUgNjExIDMxOCA0MDEgNjM2IDYzNiA2MzYgNjM2IDMzNwo1MDAgNTAwIDEwMDAgNDcxIDYxNyA4MzggMzYxIDEwMDAgNTAwIDUwMCA4MzggNDAxIDQwMSA1MDAgNjM2IDYzNiAzMTggNTAwCjQwMSA0NzEgNjE3IDk2OSA5NjkgOTY5IDUzMSA2ODQgNjg0IDY4NCA2ODQgNjg0IDY4NCA5NzQgNjk4IDYzMiA2MzIgNjMyIDYzMgoyOTUgMjk1IDI5NSAyOTUgNzc1IDc0OCA3ODcgNzg3IDc4NyA3ODcgNzg3IDgzOCA3ODcgNzMyIDczMiA3MzIgNzMyIDYxMSA2MDgKNjMwIDYxMyA2MTMgNjEzIDYxMyA2MTMgNjEzIDk5NSA1NTAgNjE1IDYxNSA2MTUgNjE1IDI3OCAyNzggMjc4IDI3OCA2MTIgNjM0CjYxMiA2MTIgNjEyIDYxMiA2MTIgODM4IDYxMiA2MzQgNjM0IDYzNCA2MzQgNTkyIDYzNSA1OTIgXQplbmRvYmoKMTUgMCBvYmoKPDwgL3QgMTYgMCBSID4+CmVuZG9iagoyMSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM5MiA+PgpzdHJlYW0KeJw9UktuBTEI288puECl8E1ynqne7t1/W5vMVKoKLwO2MZSXDKklP+qSiDNMfvVyXeJR8r1samfmIe4uNqb4WHJfuobYctGaYrFPHMkvyLRUWKFW3aND8YUoEw8ALeCBBeG+HP/xF6jB17CFcsN7ZAJgStRuQMZD0RlIWUERYfuRFeikUK9s4e8oIFfUrIWhdGKIDZYAKb6rDYmYqNmgh4SVkqod0vGMpPBbwV2JYVBbW9sEeGbQENnekY0RM+3RGXFZEWs/PemjUTK1URkPTWd88d0yUvPRFeik0sjdykNnz0InYCTmSZjncCPhnttBCzH0ca+WT2z3mClWkfAFO8oBA7393pKNz3vgLIxc2+xMJ/DRaaccE62+HmL9gz9sS5tcxyuHRRSovCgIftdBE3F8WMX3ZKNEd7QB1iMT1WglEAwSws7tMPJ4xnnZ3hW05vREaKNEHtSOET0ossXlnBWwp/yszbEcng8me2+0j5TMzKiEFdR2eqi2z2Md1Hee+/r8AS4AoRkKZW5kc3RyZWFtCmVuZG9iagoyMiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDkwID4+CnN0cmVhbQp4nE2NQRLAIAgD77wiT1BE0P90etL/X6vUDr3ATgKJFkWC9DVqSzDuuDIVa1ApmJSXwFUwXAva7qLK/jJJTJ2G03u3A4Oy8XGD0kn79nF6AKv9egbdD9IcIlgKZW5kc3RyZWFtCmVuZG9iagoyMyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE2MyA+PgpzdHJlYW0KeJxFkLl1BDEMQ3NVgRJ4gDrqGT9Hs/2nC2m83kD6eIR4iD0Jw3JdxYXRDT/etsw0vI4y3I31Zcb4qLFATtAHGCITV6NJ9e2KM1Tp4dVirqOiXC86IhLMkuOrQCN8OrLHQ1vbmX46r3/sIe8T/yoq525hAS6q7kD5Uh/x1I/ZUeqaoY8qK2seatq/CLsilLZ9XE5lnLp7B7TCZytX+30DqOc6gAplbmRzdHJlYW0KZW5kb2JqCjI0IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM3Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nOMyNDBTMDY1VcjlMjc2ArNywCwjcyMgCySLYEFk0wABXwoKCmVuZHN0cmVhbQplbmRvYmoKMjUgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA4MCA+PgpzdHJlYW0KeJxFjLsNwDAIRHumYAR+JmafKJWzfxsgStxwT7p7uDoSMlPeYYaHBJ4MLIZT8QaZo2A1uEZSjZ3so7BuX3WB5npTq/X3BypPdnZxPc3LGfQKZW5kc3RyZWFtCmVuZG9iagoyNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE0NyA+PgpzdHJlYW0KeJw9T7kNAzEM6z0FFzjAeixb81yQ6rJ/G8pGUggiQPGRZUfHClxiApOOORIvaT/4aRqBWAY1R/SEimFY4G6SAg+DLEpXni1eDJHaQl1I+NYQ3q1MZKI8rxE7cCcXowc+VBtZHnpAO0QVWa5Jw1jVVl1qnbACHLLOwnU9zKoE5dEnaykfUFRCvXT/n3va+wsAby/rCmVuZHN0cmVhbQplbmRvYmoKMjcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNDkgPj4Kc3RyZWFtCnicNY9LDgMhDEP3OYUvMFJ+hHAeqq6m9982YVoJCQvbL8EWg5GMS0xg7Jhj4SVUT60+JCOPukk5EKlQNwRPaEwMM2zSJfDKdN8ynlu8nFbqgk5I5OmsNhqijGZew9FTzgqb/svcJGplRpkDMutUtxOysmAF5gW1PPcz7qhc6ISHncqw6E4xotxmRhp+/9v0/gJ7MjBjCmVuZHN0cmVhbQplbmRvYmoKMjggMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA0OSA+PgpzdHJlYW0KeJwzNrRQMFAwNDAHkkaGQJaRiUKKIRdIAMTM5YIJ5oBZBkAaojgHriaHKw0AxugNJgplbmRzdHJlYW0KZW5kb2JqCjI5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzE3ID4+CnN0cmVhbQp4nDVSS3JDMQjbv1Nwgc6Yv32edLJq7r+thCcrsC1AQi4vWdJLftQl26XD5Fcf9yWxQj6P7ZrMUsX3FrMUzy2vR88Rty0KBFETPfgyJxUi1M/U6Dp4YZc+A68QTikWeAeTAAav4V94lE6DwDsbMt4Rk5EaECTBmkuLTUiUPUn8K+X1pJU0dH4mK3P5e3KpFGqjyQgVIFi52AekKykeJBM9iUiycr03VojekFeSx2clJhkQ3SaxTbTA49yVtISZmEIF5liA1XSzuvocTFjjsITxKmEW1YNNnjWphGa0jmNkw3j3wkyJhYbDElCbfZUJqpeP09wJI6ZHTXbtwrJbNu8hRKP5MyyUwccoJAGHTmMkCtKwgBGBOb2wir3mCzkWwIhlnZosDG1oJbt6joXA0JyzpWHG157X8/4HRVt7owplbmRzdHJlYW0KZW5kb2JqCjMwIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTMxID4+CnN0cmVhbQp4nEWPyw0EIQxD71ThEvIZPqmH1Z7Y/q/rMJpBQvhBIjvxMAis8/I20MXw0aLDN/421atjlSwfunpSVg/pkIe88hVQaTBRxIVZTB1DYc6YysiWMrcb4bZNg6xslVStg3Y8Bg+2p2WrCH6pbWHqLPEMwlVeuMcNP5BLrXe9Vb5/QlMwlwplbmRzdHJlYW0KZW5kb2JqCjMxIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjQ4ID4+CnN0cmVhbQp4nC1ROZIDQQjL5xV6QnPT77HLkff/6QrKAYOGQyA6LXFQxk8Qlive8shVtOHvmRjBd8Gh38p1GxY5EBVI0hhUTahdvB69B3YcZgLzpDUsgxnrAz9jCjd6cXhMxtntdRk1BHvXa09mUDIrF3HJxAVTddjImcNPpowL7VzPDci5EdZlGKSblcaMhCNNIVJIoeomqTNBkASjq1GjjRzFfunLI51hVSNqDPtcS9vXcxPOGjQ7Fqs8OaVHV5zLycULKwf9vM3ARVQaqzwQEnC/20P9nOzkN97SubPF9Phec7K8MBVY8ea1G5BNtfg3L+L4PePr+fwDqKVbFgplbmRzdHJlYW0KZW5kb2JqCjMyIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjEwID4+CnN0cmVhbQp4nDVQyw1DMQi7ZwoWqBQCgWSeVr11/2tt0DthEf9CWMiUCHmpyc4p6Us+OkwPti6/sSILrXUl7MqaIJ4r76GZsrHR2OJgcBomXoAWN2DoaY0aNXThgqYulUKBxSXwmXx1e+i+Txl4ahlydgQRQ8lgCWq6Fk1YtDyfkE4B4v9+w+4t5KGS88qeG/kbnO3wO7Nu4SdqdiLRchUy1LM0xxgIE0UePHlFpnDis9Z31TQS1GYLTpYBrk4/jA4AYCJeWYDsrkQ5S9KOpZ9vvMf3D0AAU7QKZW5kc3RyZWFtCmVuZG9iagoxOSAwIG9iago8PCAvQmFzZUZvbnQgL0RlamFWdVNhbnMgL0NoYXJQcm9jcyAyMCAwIFIKL0VuY29kaW5nIDw8Ci9EaWZmZXJlbmNlcyBbIDQwIC9wYXJlbmxlZnQgL3BhcmVucmlnaHQgNDYgL3BlcmlvZCA0OCAvemVybyAvb25lIC90d28gNTIgL2ZvdXIgNTQgL3NpeAo1NiAvZWlnaHQgMTA0IC9oIDExNiAvdCBdCi9UeXBlIC9FbmNvZGluZyA+PgovRmlyc3RDaGFyIDAgL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRm9udERlc2NyaXB0b3IgMTggMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUgL05hbWUgL0RlamFWdVNhbnMKL1N1YnR5cGUgL1R5cGUzIC9UeXBlIC9Gb250IC9XaWR0aHMgMTcgMCBSID4+CmVuZG9iagoxOCAwIG9iago8PCAvQXNjZW50IDkyOSAvQ2FwSGVpZ2h0IDAgL0Rlc2NlbnQgLTIzNiAvRmxhZ3MgMzIKL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRm9udE5hbWUgL0RlamFWdVNhbnMgL0l0YWxpY0FuZ2xlIDAKL01heFdpZHRoIDEzNDIgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjE3IDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDgzOCA2MDAgNjM2IDYwMCAzMTgKMzUyIDUxOCAxMDAwIDUwMCA1MDAgNTAwIDEzNDIgNjM1IDQwMCAxMDcwIDYwMCA2ODUgNjAwIDYwMCAzMTggMzE4IDUxOCA1MTgKNTkwIDUwMCAxMDAwIDUwMCAxMDAwIDUyMSA0MDAgMTAyMyA2MDAgNTI1IDYxMSAzMTggNDAxIDYzNiA2MzYgNjM2IDYzNiAzMzcKNTAwIDUwMCAxMDAwIDQ3MSA2MTIgODM4IDM2MSAxMDAwIDUwMCA1MDAgODM4IDQwMSA0MDEgNTAwIDYzNiA2MzYgMzE4IDUwMAo0MDEgNDcxIDYxMiA5NjkgOTY5IDk2OSA1MzEgNjg0IDY4NCA2ODQgNjg0IDY4NCA2ODQgOTc0IDY5OCA2MzIgNjMyIDYzMiA2MzIKMjk1IDI5NSAyOTUgMjk1IDc3NSA3NDggNzg3IDc4NyA3ODcgNzg3IDc4NyA4MzggNzg3IDczMiA3MzIgNzMyIDczMiA2MTEgNjA1CjYzMCA2MTMgNjEzIDYxMyA2MTMgNjEzIDYxMyA5ODIgNTUwIDYxNSA2MTUgNjE1IDYxNSAyNzggMjc4IDI3OCAyNzggNjEyIDYzNAo2MTIgNjEyIDYxMiA2MTIgNjEyIDgzOCA2MTIgNjM0IDYzNCA2MzQgNjM0IDU5MiA2MzUgNTkyIF0KZW5kb2JqCjIwIDAgb2JqCjw8IC9laWdodCAyMSAwIFIgL2ZvdXIgMjIgMCBSIC9oIDIzIDAgUiAvb25lIDI1IDAgUiAvcGFyZW5sZWZ0IDI2IDAgUgovcGFyZW5yaWdodCAyNyAwIFIgL3BlcmlvZCAyOCAwIFIgL3NpeCAyOSAwIFIgL3QgMzAgMCBSIC90d28gMzEgMCBSCi96ZXJvIDMyIDAgUiA+PgplbmRvYmoKMyAwIG9iago8PCAvRjEgMTkgMCBSIC9GMiAxNCAwIFIgPj4KZW5kb2JqCjQgMCBvYmoKPDwgL0ExIDw8IC9DQSAwIC9UeXBlIC9FeHRHU3RhdGUgL2NhIDEgPj4KL0EyIDw8IC9DQSAxIC9UeXBlIC9FeHRHU3RhdGUgL2NhIDEgPj4gPj4KZW5kb2JqCjUgMCBvYmoKPDwgPj4KZW5kb2JqCjYgMCBvYmoKPDwgPj4KZW5kb2JqCjcgMCBvYmoKPDwgL0RlamFWdVNhbnMtbWludXMgMjQgMCBSID4+CmVuZG9iagoyIDAgb2JqCjw8IC9Db3VudCAxIC9LaWRzIFsgMTAgMCBSIF0gL1R5cGUgL1BhZ2VzID4+CmVuZG9iagozMyAwIG9iago8PCAvQ3JlYXRpb25EYXRlIChEOjIwMTkwNTI5MDkzNzA1KzAyJzAwJykKL0NyZWF0b3IgKG1hdHBsb3RsaWIgMy4wLjMsIGh0dHA6Ly9tYXRwbG90bGliLm9yZykKL1Byb2R1Y2VyIChtYXRwbG90bGliIHBkZiBiYWNrZW5kIDMuMC4zKSA+PgplbmRvYmoKeHJlZgowIDM0CjAwMDAwMDAwMDAgNjU1MzUgZiAKMDAwMDAwMDAxNiAwMDAwMCBuIAowMDAwMDA5NzE2IDAwMDAwIG4gCjAwMDAwMDk0ODYgMDAwMDAgbiAKMDAwMDAwOTUyOSAwMDAwMCBuIAowMDAwMDA5NjI4IDAwMDAwIG4gCjAwMDAwMDk2NDkgMDAwMDAgbiAKMDAwMDAwOTY3MCAwMDAwMCBuIAowMDAwMDAwMDY1IDAwMDAwIG4gCjAwMDAwMDAzOTggMDAwMDAgbiAKMDAwMDAwMDIwOCAwMDAwMCBuIAowMDAwMDAyODQ2IDAwMDAwIG4gCjAwMDAwMDM2NDIgMDAwMDAgbiAKMDAwMDAwMzQzNCAwMDAwMCBuIAowMDAwMDAzMTE4IDAwMDAwIG4gCjAwMDAwMDQ2OTUgMDAwMDAgbiAKMDAwMDAwMjg2NyAwMDAwMCBuIAowMDAwMDA4MjYzIDAwMDAwIG4gCjAwMDAwMDgwNjMgMDAwMDAgbiAKMDAwMDAwNzY3MyAwMDAwMCBuIAowMDAwMDA5MzE2IDAwMDAwIG4gCjAwMDAwMDQ3MjcgMDAwMDAgbiAKMDAwMDAwNTE5MiAwMDAwMCBuIAowMDAwMDA1MzU0IDAwMDAwIG4gCjAwMDAwMDU1OTAgMDAwMDAgbiAKMDAwMDAwNTc2MCAwMDAwMCBuIAowMDAwMDA1OTEyIDAwMDAwIG4gCjAwMDAwMDYxMzIgMDAwMDAgbiAKMDAwMDAwNjM1NCAwMDAwMCBuIAowMDAwMDA2NDc1IDAwMDAwIG4gCjAwMDAwMDY4NjUgMDAwMDAgbiAKMDAwMDAwNzA2OSAwMDAwMCBuIAowMDAwMDA3MzkwIDAwMDAwIG4gCjAwMDAwMDk3NzYgMDAwMDAgbiAKdHJhaWxlcgo8PCAvSW5mbyAzMyAwIFIgL1Jvb3QgMSAwIFIgL1NpemUgMzQgPj4Kc3RhcnR4cmVmCjk5MzAKJSVFT0YK\n",
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (https://matplotlib.org/) -->\n",
       "<svg height=\"293.042469pt\" version=\"1.1\" viewBox=\"0 0 422.843437 293.042469\" width=\"422.843437pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 293.042469 \n",
       "L 422.843437 293.042469 \n",
       "L 422.843437 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 52.160938 282.342469 \n",
       "L 409.280938 282.342469 \n",
       "L 409.280938 16.230469 \n",
       "L 52.160938 16.230469 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" id=\"m7429de9d1d\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m7429de9d1d\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0 -->\n",
       "      <defs>\n",
       "       <path d=\"M 31.78125 66.40625 \n",
       "Q 24.171875 66.40625 20.328125 58.90625 \n",
       "Q 16.5 51.421875 16.5 36.375 \n",
       "Q 16.5 21.390625 20.328125 13.890625 \n",
       "Q 24.171875 6.390625 31.78125 6.390625 \n",
       "Q 39.453125 6.390625 43.28125 13.890625 \n",
       "Q 47.125 21.390625 47.125 36.375 \n",
       "Q 47.125 51.421875 43.28125 58.90625 \n",
       "Q 39.453125 66.40625 31.78125 66.40625 \n",
       "z\n",
       "M 31.78125 74.21875 \n",
       "Q 44.046875 74.21875 50.515625 64.515625 \n",
       "Q 56.984375 54.828125 56.984375 36.375 \n",
       "Q 56.984375 17.96875 50.515625 8.265625 \n",
       "Q 44.046875 -1.421875 31.78125 -1.421875 \n",
       "Q 19.53125 -1.421875 13.0625 8.265625 \n",
       "Q 6.59375 17.96875 6.59375 36.375 \n",
       "Q 6.59375 54.828125 13.0625 64.515625 \n",
       "Q 19.53125 74.21875 31.78125 74.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-48\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(48.979688 243.225232)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"123.584938\" xlink:href=\"#m7429de9d1d\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 2 -->\n",
       "      <defs>\n",
       "       <path d=\"M 19.1875 8.296875 \n",
       "L 53.609375 8.296875 \n",
       "L 53.609375 0 \n",
       "L 7.328125 0 \n",
       "L 7.328125 8.296875 \n",
       "Q 12.9375 14.109375 22.625 23.890625 \n",
       "Q 32.328125 33.6875 34.8125 36.53125 \n",
       "Q 39.546875 41.84375 41.421875 45.53125 \n",
       "Q 43.3125 49.21875 43.3125 52.78125 \n",
       "Q 43.3125 58.59375 39.234375 62.25 \n",
       "Q 35.15625 65.921875 28.609375 65.921875 \n",
       "Q 23.96875 65.921875 18.8125 64.3125 \n",
       "Q 13.671875 62.703125 7.8125 59.421875 \n",
       "L 7.8125 69.390625 \n",
       "Q 13.765625 71.78125 18.9375 73 \n",
       "Q 24.125 74.21875 28.421875 74.21875 \n",
       "Q 39.75 74.21875 46.484375 68.546875 \n",
       "Q 53.21875 62.890625 53.21875 53.421875 \n",
       "Q 53.21875 48.921875 51.53125 44.890625 \n",
       "Q 49.859375 40.875 45.40625 35.40625 \n",
       "Q 44.1875 33.984375 37.640625 27.21875 \n",
       "Q 31.109375 20.453125 19.1875 8.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-50\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(120.403688 243.225232)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"195.008938\" xlink:href=\"#m7429de9d1d\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 4 -->\n",
       "      <defs>\n",
       "       <path d=\"M 37.796875 64.3125 \n",
       "L 12.890625 25.390625 \n",
       "L 37.796875 25.390625 \n",
       "z\n",
       "M 35.203125 72.90625 \n",
       "L 47.609375 72.90625 \n",
       "L 47.609375 25.390625 \n",
       "L 58.015625 25.390625 \n",
       "L 58.015625 17.1875 \n",
       "L 47.609375 17.1875 \n",
       "L 47.609375 0 \n",
       "L 37.796875 0 \n",
       "L 37.796875 17.1875 \n",
       "L 4.890625 17.1875 \n",
       "L 4.890625 26.703125 \n",
       "z\n",
       "\" id=\"DejaVuSans-52\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(191.827688 243.225232)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-52\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"266.432938\" xlink:href=\"#m7429de9d1d\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 6 -->\n",
       "      <defs>\n",
       "       <path d=\"M 33.015625 40.375 \n",
       "Q 26.375 40.375 22.484375 35.828125 \n",
       "Q 18.609375 31.296875 18.609375 23.390625 \n",
       "Q 18.609375 15.53125 22.484375 10.953125 \n",
       "Q 26.375 6.390625 33.015625 6.390625 \n",
       "Q 39.65625 6.390625 43.53125 10.953125 \n",
       "Q 47.40625 15.53125 47.40625 23.390625 \n",
       "Q 47.40625 31.296875 43.53125 35.828125 \n",
       "Q 39.65625 40.375 33.015625 40.375 \n",
       "z\n",
       "M 52.59375 71.296875 \n",
       "L 52.59375 62.3125 \n",
       "Q 48.875 64.0625 45.09375 64.984375 \n",
       "Q 41.3125 65.921875 37.59375 65.921875 \n",
       "Q 27.828125 65.921875 22.671875 59.328125 \n",
       "Q 17.53125 52.734375 16.796875 39.40625 \n",
       "Q 19.671875 43.65625 24.015625 45.921875 \n",
       "Q 28.375 48.1875 33.59375 48.1875 \n",
       "Q 44.578125 48.1875 50.953125 41.515625 \n",
       "Q 57.328125 34.859375 57.328125 23.390625 \n",
       "Q 57.328125 12.15625 50.6875 5.359375 \n",
       "Q 44.046875 -1.421875 33.015625 -1.421875 \n",
       "Q 20.359375 -1.421875 13.671875 8.265625 \n",
       "Q 6.984375 17.96875 6.984375 36.375 \n",
       "Q 6.984375 53.65625 15.1875 63.9375 \n",
       "Q 23.390625 74.21875 37.203125 74.21875 \n",
       "Q 40.921875 74.21875 44.703125 73.484375 \n",
       "Q 48.484375 72.75 52.59375 71.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-54\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(263.251688 243.225232)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-54\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"337.856938\" xlink:href=\"#m7429de9d1d\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 8 -->\n",
       "      <defs>\n",
       "       <path d=\"M 31.78125 34.625 \n",
       "Q 24.75 34.625 20.71875 30.859375 \n",
       "Q 16.703125 27.09375 16.703125 20.515625 \n",
       "Q 16.703125 13.921875 20.71875 10.15625 \n",
       "Q 24.75 6.390625 31.78125 6.390625 \n",
       "Q 38.8125 6.390625 42.859375 10.171875 \n",
       "Q 46.921875 13.96875 46.921875 20.515625 \n",
       "Q 46.921875 27.09375 42.890625 30.859375 \n",
       "Q 38.875 34.625 31.78125 34.625 \n",
       "z\n",
       "M 21.921875 38.8125 \n",
       "Q 15.578125 40.375 12.03125 44.71875 \n",
       "Q 8.5 49.078125 8.5 55.328125 \n",
       "Q 8.5 64.0625 14.71875 69.140625 \n",
       "Q 20.953125 74.21875 31.78125 74.21875 \n",
       "Q 42.671875 74.21875 48.875 69.140625 \n",
       "Q 55.078125 64.0625 55.078125 55.328125 \n",
       "Q 55.078125 49.078125 51.53125 44.71875 \n",
       "Q 48 40.375 41.703125 38.8125 \n",
       "Q 48.828125 37.15625 52.796875 32.3125 \n",
       "Q 56.78125 27.484375 56.78125 20.515625 \n",
       "Q 56.78125 9.90625 50.3125 4.234375 \n",
       "Q 43.84375 -1.421875 31.78125 -1.421875 \n",
       "Q 19.734375 -1.421875 13.25 4.234375 \n",
       "Q 6.78125 9.90625 6.78125 20.515625 \n",
       "Q 6.78125 27.484375 10.78125 32.3125 \n",
       "Q 14.796875 37.15625 21.921875 38.8125 \n",
       "z\n",
       "M 18.3125 54.390625 \n",
       "Q 18.3125 48.734375 21.84375 45.5625 \n",
       "Q 25.390625 42.390625 31.78125 42.390625 \n",
       "Q 38.140625 42.390625 41.71875 45.5625 \n",
       "Q 45.3125 48.734375 45.3125 54.390625 \n",
       "Q 45.3125 60.0625 41.71875 63.234375 \n",
       "Q 38.140625 66.40625 31.78125 66.40625 \n",
       "Q 25.390625 66.40625 21.84375 63.234375 \n",
       "Q 18.3125 60.0625 18.3125 54.390625 \n",
       "z\n",
       "\" id=\"DejaVuSans-56\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(334.675688 243.225232)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-56\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"409.280938\" xlink:href=\"#m7429de9d1d\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 10 -->\n",
       "      <defs>\n",
       "       <path d=\"M 12.40625 8.296875 \n",
       "L 28.515625 8.296875 \n",
       "L 28.515625 63.921875 \n",
       "L 10.984375 60.40625 \n",
       "L 10.984375 69.390625 \n",
       "L 28.421875 72.90625 \n",
       "L 38.28125 72.90625 \n",
       "L 38.28125 8.296875 \n",
       "L 54.390625 8.296875 \n",
       "L 54.390625 0 \n",
       "L 12.40625 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-49\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(402.918438 243.225232)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- $t$ -->\n",
       "     <defs>\n",
       "      <path d=\"M 42.28125 54.6875 \n",
       "L 40.921875 47.703125 \n",
       "L 23 47.703125 \n",
       "L 17.1875 18.015625 \n",
       "Q 16.890625 16.359375 16.75 15.234375 \n",
       "Q 16.609375 14.109375 16.609375 13.484375 \n",
       "Q 16.609375 10.359375 18.484375 8.9375 \n",
       "Q 20.359375 7.515625 24.515625 7.515625 \n",
       "L 33.59375 7.515625 \n",
       "L 32.078125 0 \n",
       "L 23.484375 0 \n",
       "Q 15.484375 0 11.546875 3.125 \n",
       "Q 7.625 6.25 7.625 12.59375 \n",
       "Q 7.625 13.71875 7.765625 15.0625 \n",
       "Q 7.90625 16.40625 8.203125 18.015625 \n",
       "L 14.015625 47.703125 \n",
       "L 6.390625 47.703125 \n",
       "L 7.8125 54.6875 \n",
       "L 15.28125 54.6875 \n",
       "L 18.3125 70.21875 \n",
       "L 27.296875 70.21875 \n",
       "L 24.3125 54.6875 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-116\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(407.280938 256.903357)scale(0.1 -0.1)\">\n",
       "      <use transform=\"translate(0 0.78125)\" xlink:href=\"#DejaVuSans-Oblique-116\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" id=\"m8b4a129723\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"259.790634\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- −0.2 -->\n",
       "      <defs>\n",
       "       <path d=\"M 10.59375 35.5 \n",
       "L 73.1875 35.5 \n",
       "L 73.1875 27.203125 \n",
       "L 10.59375 27.203125 \n",
       "z\n",
       "\" id=\"DejaVuSans-8722\"/>\n",
       "       <path d=\"M 10.6875 12.40625 \n",
       "L 21 12.40625 \n",
       "L 21 0 \n",
       "L 10.6875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-46\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(20.878125 263.589853)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-8722\"/>\n",
       "       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"228.626795\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(29.257812 232.426014)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"197.462956\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(29.257812 201.262175)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"166.299117\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(29.257812 170.098336)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-52\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"135.135278\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(29.257812 138.934497)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-54\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"103.971439\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(29.257812 107.770658)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-56\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"72.8076\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_14\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(29.257812 76.606819)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_8\">\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m8b4a129723\" y=\"41.643761\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_15\">\n",
       "      <!-- 1.2 -->\n",
       "      <g transform=\"translate(29.257812 45.44298)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_16\">\n",
       "     <!-- h(t) -->\n",
       "     <defs>\n",
       "      <path d=\"M 54.890625 33.015625 \n",
       "L 54.890625 0 \n",
       "L 45.90625 0 \n",
       "L 45.90625 32.71875 \n",
       "Q 45.90625 40.484375 42.875 44.328125 \n",
       "Q 39.84375 48.1875 33.796875 48.1875 \n",
       "Q 26.515625 48.1875 22.3125 43.546875 \n",
       "Q 18.109375 38.921875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 75.984375 \n",
       "L 18.109375 75.984375 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.34375 51.125 25.703125 53.5625 \n",
       "Q 30.078125 56 35.796875 56 \n",
       "Q 45.21875 56 50.046875 50.171875 \n",
       "Q 54.890625 44.34375 54.890625 33.015625 \n",
       "z\n",
       "\" id=\"DejaVuSans-104\"/>\n",
       "      <path d=\"M 31 75.875 \n",
       "Q 24.46875 64.65625 21.28125 53.65625 \n",
       "Q 18.109375 42.671875 18.109375 31.390625 \n",
       "Q 18.109375 20.125 21.3125 9.0625 \n",
       "Q 24.515625 -2 31 -13.1875 \n",
       "L 23.1875 -13.1875 \n",
       "Q 15.875 -1.703125 12.234375 9.375 \n",
       "Q 8.59375 20.453125 8.59375 31.390625 \n",
       "Q 8.59375 42.28125 12.203125 53.3125 \n",
       "Q 15.828125 64.359375 23.1875 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-40\"/>\n",
       "      <path d=\"M 18.3125 70.21875 \n",
       "L 18.3125 54.6875 \n",
       "L 36.8125 54.6875 \n",
       "L 36.8125 47.703125 \n",
       "L 18.3125 47.703125 \n",
       "L 18.3125 18.015625 \n",
       "Q 18.3125 11.328125 20.140625 9.421875 \n",
       "Q 21.96875 7.515625 27.59375 7.515625 \n",
       "L 36.8125 7.515625 \n",
       "L 36.8125 0 \n",
       "L 27.59375 0 \n",
       "Q 17.1875 0 13.234375 3.875 \n",
       "Q 9.28125 7.765625 9.28125 18.015625 \n",
       "L 9.28125 47.703125 \n",
       "L 2.6875 47.703125 \n",
       "L 2.6875 54.6875 \n",
       "L 9.28125 54.6875 \n",
       "L 9.28125 70.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-116\"/>\n",
       "      <path d=\"M 8.015625 75.875 \n",
       "L 15.828125 75.875 \n",
       "Q 23.140625 64.359375 26.78125 53.3125 \n",
       "Q 30.421875 42.28125 30.421875 31.390625 \n",
       "Q 30.421875 20.453125 26.78125 9.375 \n",
       "Q 23.140625 -1.703125 15.828125 -13.1875 \n",
       "L 8.015625 -13.1875 \n",
       "Q 14.5 -2 17.703125 9.0625 \n",
       "Q 20.90625 20.125 20.90625 31.390625 \n",
       "Q 20.90625 42.671875 17.703125 53.65625 \n",
       "Q 14.5 64.65625 8.015625 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-41\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(14.798438 25.260937)rotate(-90)scale(0.1 -0.1)\">\n",
       "      <use xlink:href=\"#DejaVuSans-104\"/>\n",
       "      <use x=\"63.378906\" xlink:href=\"#DejaVuSans-40\"/>\n",
       "      <use x=\"102.392578\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use x=\"141.601562\" xlink:href=\"#DejaVuSans-41\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"LineCollection_1\">\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 52.160938 228.626795 \n",
       "L 55.477442 163.068839 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 55.477442 163.068839 \n",
       "L 58.735793 111.986803 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 58.735793 111.986803 \n",
       "L 60.615515 88.441656 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 60.615515 88.441656 \n",
       "L 62.724796 67.006701 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 62.724796 67.006701 \n",
       "L 63.497025 60.434701 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 63.497025 60.434701 \n",
       "L 64.296536 54.328814 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 64.296536 54.328814 \n",
       "L 65.165603 48.478859 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 65.165603 48.478859 \n",
       "L 66.148556 42.823325 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 66.148556 42.823325 \n",
       "L 66.950868 38.940682 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 66.950868 38.940682 \n",
       "L 67.658682 36.046202 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 67.658682 36.046202 \n",
       "L 68.051999 34.647593 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 68.051999 34.647593 \n",
       "L 68.40307 33.523299 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 68.40307 33.523299 \n",
       "L 68.761537 32.494213 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 68.761537 32.494213 \n",
       "L 69.175318 31.453265 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 69.175318 31.453265 \n",
       "L 69.483405 30.778825 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 69.483405 30.778825 \n",
       "L 69.798702 30.176112 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 69.798702 30.176112 \n",
       "L 70.099699 29.682121 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 70.099699 29.682121 \n",
       "L 70.438269 29.220012 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 70.438269 29.220012 \n",
       "L 70.736806 28.893443 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 70.736806 28.893443 \n",
       "L 71.085115 28.606668 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 71.085115 28.606668 \n",
       "L 71.401425 28.432712 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 71.401425 28.432712 \n",
       "L 71.766697 28.332378 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 71.766697 28.332378 \n",
       "L 72.038737 28.326469 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 72.038737 28.326469 \n",
       "L 72.276602 28.368666 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 72.276602 28.368666 \n",
       "L 72.699995 28.551185 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 72.699995 28.551185 \n",
       "L 73.110067 28.85638 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 73.110067 28.85638 \n",
       "L 73.463912 29.218912 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 73.463912 29.218912 \n",
       "L 73.87575 29.753812 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 73.87575 29.753812 \n",
       "L 74.369707 30.551425 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 74.369707 30.551425 \n",
       "L 74.747694 31.27357 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 74.747694 31.27357 \n",
       "L 75.169088 32.189645 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 75.169088 32.189645 \n",
       "L 75.74154 33.615733 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 75.74154 33.615733 \n",
       "L 76.438404 35.622944 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 76.438404 35.622944 \n",
       "L 77.178482 38.064691 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 77.178482 38.064691 \n",
       "L 77.962926 40.981713 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 77.962926 40.981713 \n",
       "L 79.39169 47.095315 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 79.39169 47.095315 \n",
       "L 80.566729 52.822157 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 80.566729 52.822157 \n",
       "L 83.366545 68.581379 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 83.366545 68.581379 \n",
       "L 88.550306 103.077989 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 88.550306 103.077989 \n",
       "L 94.341572 144.597046 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 94.341572 144.597046 \n",
       "L 99.271509 178.400784 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 99.271509 178.400784 \n",
       "L 103.919271 206.630675 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 103.919271 206.630675 \n",
       "L 106.402108 219.781014 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 106.402108 219.781014 \n",
       "L 108.814637 231.11521 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 108.814637 231.11521 \n",
       "L 111.154897 240.690829 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 111.154897 240.690829 \n",
       "L 112.507485 245.57831 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 112.507485 245.57831 \n",
       "L 113.74966 249.649582 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 113.74966 249.649582 \n",
       "L 115.138444 253.732264 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 115.138444 253.732264 \n",
       "L 116.666443 257.660718 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 116.666443 257.660718 \n",
       "L 117.954845 260.524339 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 117.954845 260.524339 \n",
       "L 119.08267 262.702726 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 119.08267 262.702726 \n",
       "L 120.537244 265.073583 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 120.537244 265.073583 \n",
       "L 121.787595 266.730629 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 121.787595 266.730629 \n",
       "L 122.917025 267.936889 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 122.917025 267.936889 \n",
       "L 124.17996 268.973194 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 124.17996 268.973194 \n",
       "L 125.668203 269.792369 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 125.668203 269.792369 \n",
       "L 127.182551 270.207058 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 127.182551 270.207058 \n",
       "L 128.519177 270.246469 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 128.519177 270.246469 \n",
       "L 129.868364 269.998762 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 129.868364 269.998762 \n",
       "L 132.184666 268.965495 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 132.184666 268.965495 \n",
       "L 134.901894 266.909565 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 134.901894 266.909565 \n",
       "L 139.402266 261.973438 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 139.402266 261.973438 \n",
       "L 145.996415 252.728698 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 145.996415 252.728698 \n",
       "L 151.606419 244.377234 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 151.606419 244.377234 \n",
       "L 158.795331 234.650844 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 158.795331 234.650844 \n",
       "L 165.532092 227.552515 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 165.532092 227.552515 \n",
       "L 171.698319 223.14725 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 171.698319 223.14725 \n",
       "L 177.22649 220.877956 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 177.22649 220.877956 \n",
       "L 183.48417 219.977835 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 183.48417 219.977835 \n",
       "L 190.755684 220.623677 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 190.755684 220.623677 \n",
       "L 196.245077 221.893758 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 196.245077 221.893758 \n",
       "L 201.912091 223.560637 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 201.912091 223.560637 \n",
       "L 208.149811 225.487803 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 208.149811 225.487803 \n",
       "L 215.131375 227.434922 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 215.131375 227.434922 \n",
       "L 221.76421 228.874865 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 221.76421 228.874865 \n",
       "L 227.707743 229.755908 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 227.707743 229.755908 \n",
       "L 234.113472 230.280175 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 234.113472 230.280175 \n",
       "L 241.530329 230.420276 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 241.530329 230.420276 \n",
       "L 246.394475 230.307386 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 246.394475 230.307386 \n",
       "L 251.429917 230.076784 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 251.429917 230.076784 \n",
       "L 257.746763 229.696736 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 257.746763 229.696736 \n",
       "L 263.067073 229.353805 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 263.067073 229.353805 \n",
       "L 269.028518 228.993751 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 269.028518 228.993751 \n",
       "L 274.821921 228.700396 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 274.821921 228.700396 \n",
       "L 282.150849 228.434809 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 282.150849 228.434809 \n",
       "L 288.63457 228.301939 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 288.63457 228.301939 \n",
       "L 293.581752 228.258492 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 293.581752 228.258492 \n",
       "L 299.040507 228.257903 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 299.040507 228.257903 \n",
       "L 304.674775 228.29587 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 304.674775 228.29587 \n",
       "L 309.354814 228.346769 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 309.354814 228.346769 \n",
       "L 313.737545 228.402971 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 313.737545 228.402971 \n",
       "L 317.568676 228.45441 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 317.568676 228.45441 \n",
       "L 322.82811 228.522814 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 322.82811 228.522814 \n",
       "L 327.499992 228.577191 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 327.499992 228.577191 \n",
       "L 332.403927 228.624762 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 332.403927 228.624762 \n",
       "L 337.604224 228.662897 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 337.604224 228.662897 \n",
       "L 342.808323 228.68812 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 342.808323 228.68812 \n",
       "L 347.591355 228.700701 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 347.591355 228.700701 \n",
       "L 351.894111 228.704513 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 351.894111 228.704513 \n",
       "L 356.932705 228.701704 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 356.932705 228.701704 \n",
       "L 360.999873 228.695139 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 360.999873 228.695139 \n",
       "L 365.5772 228.684688 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 365.5772 228.684688 \n",
       "L 371.92693 228.667489 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 371.92693 228.667489 \n",
       "L 378.539537 228.649408 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 378.539537 228.649408 \n",
       "L 384.323613 228.635501 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 384.323613 228.635501 \n",
       "L 389.265778 228.62588 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 389.265778 228.62588 \n",
       "L 394.00715 228.618907 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 394.00715 228.618907 \n",
       "L 398.850036 228.614088 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 398.850036 228.614088 \n",
       "L 404.349591 228.61122 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "    <path clip-path=\"url(#p49e5ced245)\" d=\"M 404.349591 228.61122 \n",
       "L 409.280938 228.610649 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-width:1.5;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 52.160938 270.246469 \n",
       "L 52.160938 28.326469 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 409.280938 282.342469 \n",
       "L 409.280938 16.230469 \n",
       "\" style=\"fill:none;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 52.160938 228.626795 \n",
       "L 409.280938 228.626795 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 52.160938 16.230469 \n",
       "L 409.280938 16.230469 \n",
       "\" style=\"fill:none;\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p49e5ced245\">\n",
       "   <rect height=\"266.112\" width=\"357.12\" x=\"52.160938\" y=\"16.230469\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10eafff98>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.plot(h, (t, 0, 10), xlabel='$t$', ylabel='h(t)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note, the result is equal to the [impulse response derived by explicit solution of the underlying ODE](../systems_time_domain/network_analysis.ipynb#Impulse-Response)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Relation to Differential Equations\n",
    "\n",
    "Linear ODEs with constant coefficients describing the relation between input $x(t)$ and output $y(t)$ represent an LTI system. In this case a system is characterized by the constants $\\alpha_n$ and $\\beta_m$ of the ODE\n",
    "\n",
    "\\begin{equation}\n",
    "\\sum_{n=0}^{N} \\alpha_n \\frac{d^n}{d t^n} y(t) = \\sum_{m=0}^{M} \\beta_m \\frac{d^m}{d t^m} x(t)\n",
    "\\end{equation}\n",
    "\n",
    "where $\\alpha_N \\neq 0$ with $N$ denoting the degree of the ODE. Initial values are discarded in the following. Applying a two-sided Laplace transform to the left- and right-hand-side and using the [differentiation theorem](../laplace_transform/theorems.ipynb#Differentiation-Theorem) yields\n",
    "\n",
    "\\begin{equation}\n",
    "Y(s) \\sum_{n=0}^{N} \\alpha_n s^n = X(s) \\sum_{m=0}^{M} \\beta_m s^m\n",
    "\\end{equation}\n",
    "\n",
    "Rearranging this result derives the transfer function $H(s)$ of the system as\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) = \\frac{Y(s)}{X(s)} = \\frac{\\sum_{m=0}^{M} \\beta_m s^m}{\\sum_{n=0}^{N} \\alpha_n s^n}\n",
    "\\end{equation}\n",
    "\n",
    "The transfer function of an LTI system whose in- and output relation can be characterized by an ODE is given as rational function in $s$. The coefficients of its numerator and denominator polynomial are given by the coefficients $\\beta_m$ and $\\alpha_n$ of the ODE. The conversion between both representations of an LTI system is therefore straightforward."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The coefficients for the 2nd order low-pass filter can be determined [from its ODE](../systems_time_domain/network_analysis.ipynb#Differential-Equation) or from above transfer function as\n",
    "\n",
    "\\begin{align}\n",
    "\\beta_0 &= 1 & \\alpha_0 &= 1 \\\\\n",
    "& & \\alpha_1 &= C R \\\\\n",
    "& & \\alpha_2 &= C L\n",
    "\\end{align}\n",
    "\n",
    "From theses coefficients the transfer function $H(s)$ is constructed in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAtCAYAAAC9B8nZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFyUlEQVR4Ae2ci3HVOhCGOZkUANwOQgc8OoAObu6tAOggGSpgoAOggntDB0AFDHRAOgDSQfg/ofX4Idmyj0zOSVYziqzHrrS/f61k+cSby8vLWx7KEdhsNvfV+kzxgbC7KJe8uS0Pb67p5ZaLWLfV+p3iD8WHikeKHgoRcJIVABU91jFNRbgTJXgzD4UIHBS282aOwGIEnGSLoXPBUgScZKVIebvFCDjJFkPngqUIOMlKkfJ2ixFwki2GzgVLEXCSlSLl7RYj4CRbDJ0LliLgJCtFytstRsBJthg6FyxFYOMvyMug0uskXorzDpN3l6Tnil8VPwvD10o9ZBBwkmWA8eJ6CPhyWQ9L15RBwEmWAcaL6yHgJKuHpWvKIOAkywDjxfUQcJLVw9I1ZRBwkmWA8eJ6CDjJ6mHpmjIIOMkywHhxPQQ2UuX/E1cPT9eUQMBP/BOgeFFdBHy5rIuna0sg4CRLgOJFdRHYm3/ujf/F/SKafxTTp/6pgLqEWEPb3pBMxr8SoZ4bCCLdG11/UbxnZZ7uJgLZjX/0HM807CeK7Q+LfNDNfqt6vMlzXZ/2TVMd/8r/SPHvWMfvrviOxBmysWxWIp08BT+R/EcEY//fdMmHT9B/ZUFj4fdl22AFxo+jAdgH3ui08EY2vrfMrqbCIf0xGg3+Vj/KCEjyM6a32/UqgzivFLnBz9p1/WvVQ4xv/fIlefpSbMaia0iO/vtL9NWSUf81sWICd+4H9kU7maCdul3Ia2xMBn7QaSsL96S5T4yxM+go8EEpBOo0bBukOhSi7Khd3r5WnYHDMtfpp0Ze+gPRl+qSPJ5jdJKM6ZYs4NbG6iTVZ+xnFO+UXGmZ9G+FhfUjPUy4Acn6T5ef1IifF099ewvmXkg5P0HOBXP/3IiqIbplPOqDLRRDkvaSNFfVGliFrUBiIHcTZTWLtsVidCyHVqsbh2fA+xyLPBdWnknZX/2fqbPif7mQrhxw1m5WqnGyTDLWqYkwS++cxitiNdhbxgnFffk4MannmPBH2waSxRuHqzuXISUbTDwYS+ZYAJgBaGMC8eZ9j23+UvqyTfg4TpZfNsps/uljyqPStFpYEavBZIx9sWq8l83h+2h9Q9SGCZfFrN/+KvLmyexoYIo4YYzxxmcJJMNtqRwAlzNSMiyrp9Id9EYdABjGpjwejPGd6hpyEagbPN2GmvX+rIWVfWCPkXMsg71MZlaWJNZTmEl2J4KRzI4aikkxMfrgadSmaD8WCfS4BybkYRZb4EyMvQNpEyRjN70pW/liLaw4Dgp7XOGBne8UjWgDkwoxG8hdRYGR7G7sfGwj34wPEATI2L4teDK1mUVa6YWUEIv9hxE19Kv8nWYAMy6kE+9nnrUtGWxWfYqkX9VfcnmSgjWw6iz56htsjzU2ntReKGa99RhmkuuEFbDo6M9mZBB1HFnocvqoQe2YyaNnU+hS/JLTpzrOvDrHH8qzNCJncZWjDxuT+sGO5JGBtUmlklkDq8EZmPrBm4HFoM7GpboqmEnPIixsHJZKz+gRRvA4YjrueSo8ktLkHgFB6TCv8d+IIk7uO15TefZj/L4ND8Z4TqQLoHctrIFValthOP7IAbAvmB1EA5gRBNtv/M71/uqm0+5lr7iftWUuuVRKB2xvQEWnIt4hBAE3WCqtbkfSmljZkpw6DuK1HKHBRjiFB559w+wQK3RjzzVwDD5Tyv6g835RZZAPoztHCsqnQiCqdAy8HeBIAJK191fM2M5TbezvrXSM7ftSfa9eVgMr2YeHZtX4Z2TADbloIxnzbGT3CrPOC/JoPEQAALvBuGv2BUnPpDoAADR7GgqzTfn2eRv6mnLpshmMLOVtAJUNZ18dolNYM6hfJgP7wkUfS1mKFTZI1rxh2yTe8XZsVjsmH9jh+ZvJr/KqmEnftljwsAYHHsaUrRBO5jP4dkimwhsTtgX2OgG1NhYH1wmsmbbgqc1bzxS9ds1XxeIXV3ZHxlm0mAcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{C L s^{2} + C R s + 1}$"
      ],
      "text/plain": [
       "        1         \n",
       "──────────────────\n",
       "     2            \n",
       "C⋅L⋅s  + C⋅R⋅s + 1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R, L, C = sym.symbols('R L C', positive=True)\n",
    "beta = 1\n",
    "alpha = L*C, R*C, 1\n",
    "\n",
    "H = sym.Poly(beta, s) / sym.Poly(alpha, s)\n",
    "H"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Many software packages for signal processing feature functionality for the computation of the transfer function from the coefficients $\\beta_m$ and $\\alpha_n$ of a system. For instance, the Python signal processing toolbox [`scipy.signal`](http://docs.scipy.org/doc/scipy/reference/signal.html) provides a function to numerically evaluate the transfer function $H(j \\omega)$ at a number of angular frequencies $\\omega_i$ for a given set of coefficients. This is illustrated in the following for the normalized values $R$, $L$ and $C$ used above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import signal\n",
    "\n",
    "R = 1\n",
    "L = 0.5\n",
    "C = 0.4\n",
    "beta = 1\n",
    "alpha = L*C, R*C, 1\n",
    "\n",
    "w, H = signal.freqresp((beta, alpha))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This results in the vector `w` of discrete angular frequencies $\\omega_i$ and the vector `H` with the values of $H(j \\omega)$ at these frequencies. The latter is plotted with the Python plotting library [`matplotlib`](http://matplotlib.org/)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": "JVBERi0xLjQKJazcIKu6CjEgMCBvYmoKPDwgL1BhZ2VzIDIgMCBSIC9UeXBlIC9DYXRhbG9nID4+CmVuZG9iago4IDAgb2JqCjw8IC9FeHRHU3RhdGUgNCAwIFIgL0ZvbnQgMyAwIFIgL1BhdHRlcm4gNSAwIFIKL1Byb2NTZXQgWyAvUERGIC9UZXh0IC9JbWFnZUIgL0ltYWdlQyAvSW1hZ2VJIF0gL1NoYWRpbmcgNiAwIFIKL1hPYmplY3QgNyAwIFIgPj4KZW5kb2JqCjEwIDAgb2JqCjw8IC9Bbm5vdHMgWyBdIC9Db250ZW50cyA5IDAgUgovR3JvdXAgPDwgL0NTIC9EZXZpY2VSR0IgL1MgL1RyYW5zcGFyZW5jeSAvVHlwZSAvR3JvdXAgPj4KL01lZGlhQm94IFsgMCAwIDQxMi45MjYyNSAzMTQuNjgzODc1IF0gL1BhcmVudCAyIDAgUiAvUmVzb3VyY2VzIDggMCBSCi9UeXBlIC9QYWdlID4+CmVuZG9iago5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTEgMCBSID4+CnN0cmVhbQp4nMWZS49cxw2F9/dX1FJeqFRkvZcWHCvIJpA9SBZJFrI8VjTQyFAGijf58fnIO/2442mP3AE6AoTpZtdlkYeHrFPdEm4WCe9CCjf8/yX8LfyDvz8GCa/4/25JvLtdimic2rTy7sPRuywltpFHr5jT9u0/l+Wn5cXXOLnjqVfLUmqU5E/1OLrYKjwnjapb64dja8px7lwePBxbfaNP4RH3ufYoGrS1KPz913X4a/gYXnytFpOGP5EhWcdxyHtJsc0ks6U+ChC8+Ob63+/fXn/36mV4e7fUEVOdLY9NCkfW47CW75fX4dNutwTGv9rNcLFPnnZcYt5hs7ykYL8sL6/Ci28lSApXPy218NScNU2dI2iO2ocAxdWPy7P0Vbi6CX+48mguAJKwfSutyLbQx+ZzYfoC108AJZpiaa1r72PmB0jppZEaM2rOMnWbzpH5bKSedv0UUqPBvlwTwI6HnCoXRkqN4DJnm5t0js3nIvUFrp9ASqtAv6RDahoPOdUujFQ2gqfWS9ukc2w+F6kvcP0EUlk6Y72nOnLrDzk1Lo3UqHHOXDVv0zkyn43U066fQqrP2HKR0hlVDzkl+5H+iSdTeJ6IRQlWmkE2d2vf3uJ0GewEO3PJ7JZigqptdXTY7lkwjwseOtGm6Ut9r/f+gUbmJYXOorsPPvoHLebMJE1tbz/l6cW3utvszpeIxJpnTnPMygkXh+4StLi6rxRfaYla5JZr2r/YZ0KaVPbmzV8+f//m493zP//w4f2nz9fPf769fvcmfPPz8np9OvXd47zSGatKh5A9+eZTvRoPfN2+//j5bvXx+lK8PGxBjJ1RPbcyaW/9fZx83C2TK/3K62Mzju3JFBrUVEuCw6u2iJdWF/staJBZtAzZoHOwnonOxu0OnY3XU+h0Miy1JXDco3NpRbHfwnROFgaYbuA5Mp+Jz9bxDqCt31MIibRYpUnteVa5h+jSUuKQSUt2Wams2UB0MJ8L0cbxHqKN35MQ1RbHaD3N1PcQXVpD7LfQJGjAIrltIDoynwnR1vEOoq3fkxBNRETtSux1D9GlxcMhE6JnNLTRtxAdzOdCtHG8h2jj9xRESuLNRUDJ95Lh/zeoOdZjn4ie7Tl2ZD4XoY3jPUIbvycR6h0phgwtIjuE9KCqTFA9N5Uh1ZsSHQicfSepTEXk6tpmK6P+cy9+cptSDqLoSPv80Vd05uCsBzV15OLvz+59AC0q2Wvz0MnNTkBx0tzruodevjrh5T7ERyQV2rzagFol2y6/p8XV/yaMEIEbRokyAgfktRi4qa//kG7Rrg1DUR4zfPeQbEsThOiYvXuH8ATnNaxoGVmTcrPGUQIc06Zvq4hNrbm4tQ+dxXR4G6xgunAxKRlqj2qLO2GonepY7VuWsh5ynQfnSLYWDay1qnVkr369FvM8YupTqtEQsqGbSsdcSZGIkq0ehjLbE11VGhbO2OqRPY6GtaJSB4LCrFzH8SbEUUekNlwOzIzah8IJ1ySbtE216Tk1zsaGxEdamjqYYpZUKHIZmSvF4KVkzeJ2jp0krbLp6DFx7RV1+4it2hDRwaZQSbKZBWnT8my4mSQE2GO103sGIVWfjJxG8Kud0BJ6u7q9ztIdRIAitlLqun6gLftqr2DaCz3s9jZK9TAREEWmuIC3MlsZ3Q53U1m9aKkrBiIEnybD0qOsdoVZvczYGy2Kd+BrKBJxL4iY2cb0XLOweLiVeVG53rInNaizs97tcATNsyKZocK6qdII2meZPomqX5rc3uAA5a7r7CkVuN0+YqUgBT89xwIz1CHLREzzViss05y7jisHKkbxlYJhN/7UOd0P4nxItiCMY4QJOGYvNDdeG/GwJMNZ6W63AVimAhrF743e8PUVvEvJWjhouRfZXd33hYtUpA2UDASlyGzhduKvNAzrPf4pazzNqdCJX1qBOmmu/nkzoMKwscp8KF2n50uSdKcUBlG1O1qiCman/wzyNIPQlYV57c0qvcQ0ZHB1FEZOpvv9TORCxwHDsw07FIHt3eOxZpsK9YKVIpPjWnSSYa5wh8XO2co9MHk81Np4N121jp5bczNEUtuLMEVpID7wMMGKQ1nglPVMId2y2gmHmlpa1nuwNxkdNHG4wV8mm9hLJpd3BN1ArWuZI/Ckfa3iDaf4rM12C7PAr0xjmZkQaEN0UCCuOSjQam7ukTU0M+bpFFTyy8XehWEkpQ/cN3AANw08FIhKWV1nC7D63adDFB9vmlmai41Lg5m/LtVs5DL/ONACVRmcIU4xhWIcsAYt07IyUhwqBi9vSJcrkxE4F+OXkdnGGz5sYHH0JMcDWiAX0pRgx8iszbtGbR8pVCLAJ9Rg955UMoDTlcZtCnrpPj6bIfQhhDIRToepBZKRm7kDsASbqMwhn6jZhhiBAAMEVSbF9NX2HUwTG/m0QqfHqwWY7VLcLLNKIBm6GaZgBPUq84vBPmC7Y8o5xGhlyLMhVKMNfGTQtDxpTR1gPlNyHaZH3xbhbba+flt0wrxVUHa6ruop/cb3C8c/wzz6ywpOH/t95vbU7zO2/st/5NmsPrj5Le+vl/8Cz2mJfAplbmRzdHJlYW0KZW5kb2JqCjExIDAgb2JqCjE5MjUKZW5kb2JqCjE2IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggODggPj4Kc3RyZWFtCnicPY25EcAwCARzqqAEgXikfjyO5P5Tn/ATADvs3Fy6cmNNrFTHTD6EJGbhRRqjaJFKY+sN5Pg9FBqv/cKwo/N2Nqyuo6FbgmTaR8htV73rr1t03skLG84KZW5kc3RyZWFtCmVuZG9iagoxNyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE1MiA+PgpzdHJlYW0KeJw9T8sRQyEIvFvFNsCMIILW8zI5mf6vAU08sbK4H/GBCmIWkFSHuMOt4sWFtaOr41OkHbQKi4PmBpa/ErGCvIHr1ukYPWabeIrnxhg6Y4awcMyRDFnaxTPuti9Fjg2Cu2FWoekplj7kemEbB1J6s7RdoaZwLqYiY4Zx22mfWyXSjtMvSv2ariK9w9SStYvGn31/Abz3NmcKZW5kc3RyZWFtCmVuZG9iagoxOCAwIG9iago8PCAvQkJveCBbIC0xMDE2IC0zNTEgMTY2MCAxMDY4IF0gL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxODEKL1N1YnR5cGUgL0Zvcm0gL1R5cGUgL1hPYmplY3QgPj4Kc3RyZWFtCnicNVA5DgQhDOt5hT+wErnDe0baavb/7YYwU6AYHAc7g0nxIcFvRDaQiRR4gsNxjWSIMYgNqgJaDtMogpc0uge/XGr1EihmiaV6SEspdVzhDJbY5Roy+y6mIF2QDEgNuYaWdKN7qBxObcIJRnmktuYGrrMml882uv/ydkrwiHbjhY/T0HycRmXcXKR1qgh6YkbNyj02eglG3LUcsZy11FKkVIs6UWWw1Q9zNv/u8RrfP+Y3Px4KZW5kc3RyZWFtCmVuZG9iagoxOSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDMyNSA+PgpzdHJlYW0KeJw1krlxQzEQQ/NfxTbgGe7Fox55HNn9p36g7EADiOSCAD473Ia524en9RjWc9inP5em289TO61q2vdTfax6WyUrExzLarW9ntxuxS+7wLaMefH1xNqXRXFiLQvnxHR2fDNdbb5YibyYRzOXzbQYKOW2mAdFv2pp6ccCpXR21rmIg7Etx8Fl4ixO4E770gZ7cqYKhp4yBHGFvkM7YsHZbJP0Jqjuy6IUC86onkBRKIfnMi+h+lv2IeskHcbiVqUxZPlAE1+EJt0pQ83pB2sIcfGkgw0w4BcYmnH/n2VrkL0NR9hpRhV0TvOMN96BywidOVChZkXMJSncRKoYqUctmK/gBMbyXZHsx5GfM+kTiwAzgqUQENXofx/c23krcuREakoKPrxeju4XytGbqctu5rij+5ZdmpFSgMP+n9rr+foFmDF5uwplbmRzdHJlYW0KZW5kb2JqCjE0IDAgb2JqCjw8IC9CYXNlRm9udCAvRGVqYVZ1U2Fucy1PYmxpcXVlIC9DaGFyUHJvY3MgMTUgMCBSCi9FbmNvZGluZyA8PCAvRGlmZmVyZW5jZXMgWyA3MiAvSCAxMDYgL2ogMTE1IC9zIF0gL1R5cGUgL0VuY29kaW5nID4+Ci9GaXJzdENoYXIgMCAvRm9udEJCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9Gb250RGVzY3JpcHRvciAxMyAwIFIKL0ZvbnRNYXRyaXggWyAwLjAwMSAwIDAgMC4wMDEgMCAwIF0gL0xhc3RDaGFyIDI1NSAvTmFtZSAvRGVqYVZ1U2Fucy1PYmxpcXVlCi9TdWJ0eXBlIC9UeXBlMyAvVHlwZSAvRm9udCAvV2lkdGhzIDEyIDAgUiA+PgplbmRvYmoKMTMgMCBvYmoKPDwgL0FzY2VudCA5MjkgL0NhcEhlaWdodCAwIC9EZXNjZW50IC0yMzYgL0ZsYWdzIDk2Ci9Gb250QkJveCBbIC0xMDE2IC0zNTEgMTY2MCAxMDY4IF0gL0ZvbnROYW1lIC9EZWphVnVTYW5zLU9ibGlxdWUKL0l0YWxpY0FuZ2xlIDAgL01heFdpZHRoIDEzNTAgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjEyIDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDgzOCA2MDAgNjM2IDYwMCAzMTgKMzUyIDUxOCAxMDAwIDUwMCA1MDAgNTAwIDEzNTAgNjM1IDQwMCAxMDcwIDYwMCA2ODUgNjAwIDYwMCAzMTggMzE4IDUxOCA1MTgKNTkwIDUwMCAxMDAwIDUwMCAxMDAwIDUyMSA0MDAgMTAyOCA2MDAgNTI1IDYxMSAzMTggNDAxIDYzNiA2MzYgNjM2IDYzNiAzMzcKNTAwIDUwMCAxMDAwIDQ3MSA2MTcgODM4IDM2MSAxMDAwIDUwMCA1MDAgODM4IDQwMSA0MDEgNTAwIDYzNiA2MzYgMzE4IDUwMAo0MDEgNDcxIDYxNyA5NjkgOTY5IDk2OSA1MzEgNjg0IDY4NCA2ODQgNjg0IDY4NCA2ODQgOTc0IDY5OCA2MzIgNjMyIDYzMiA2MzIKMjk1IDI5NSAyOTUgMjk1IDc3NSA3NDggNzg3IDc4NyA3ODcgNzg3IDc4NyA4MzggNzg3IDczMiA3MzIgNzMyIDczMiA2MTEgNjA4CjYzMCA2MTMgNjEzIDYxMyA2MTMgNjEzIDYxMyA5OTUgNTUwIDYxNSA2MTUgNjE1IDYxNSAyNzggMjc4IDI3OCAyNzggNjEyIDYzNAo2MTIgNjEyIDYxMiA2MTIgNjEyIDgzOCA2MTIgNjM0IDYzNCA2MzQgNjM0IDU5MiA2MzUgNTkyIF0KZW5kb2JqCjE1IDAgb2JqCjw8IC9IIDE2IDAgUiAvaiAxNyAwIFIgL3MgMTkgMCBSID4+CmVuZG9iagoyNCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDU0ID4+CnN0cmVhbQp4nDM2NlcwUDA0MlfQNTI2VTAyNFAwNzNRSDHkgjFzwSywbA4XXCGECZLPgavM4UoDAEyQDxUKZW5kc3RyZWFtCmVuZG9iagoyNSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM5MiA+PgpzdHJlYW0KeJw9UktuBTEI288puECl8E1ynqne7t1/W5vMVKoKLwO2MZSXDKklP+qSiDNMfvVyXeJR8r1samfmIe4uNqb4WHJfuobYctGaYrFPHMkvyLRUWKFW3aND8YUoEw8ALeCBBeG+HP/xF6jB17CFcsN7ZAJgStRuQMZD0RlIWUERYfuRFeikUK9s4e8oIFfUrIWhdGKIDZYAKb6rDYmYqNmgh4SVkqod0vGMpPBbwV2JYVBbW9sEeGbQENnekY0RM+3RGXFZEWs/PemjUTK1URkPTWd88d0yUvPRFeik0sjdykNnz0InYCTmSZjncCPhnttBCzH0ca+WT2z3mClWkfAFO8oBA7393pKNz3vgLIxc2+xMJ/DRaaccE62+HmL9gz9sS5tcxyuHRRSovCgIftdBE3F8WMX3ZKNEd7QB1iMT1WglEAwSws7tMPJ4xnnZ3hW05vREaKNEHtSOET0ossXlnBWwp/yszbEcng8me2+0j5TMzKiEFdR2eqi2z2Md1Hee+/r8AS4AoRkKZW5kc3RyZWFtCmVuZG9iagoyNiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDkwID4+CnN0cmVhbQp4nE2NQRLAIAgD77wiT1BE0P90etL/X6vUDr3ATgKJFkWC9DVqSzDuuDIVa1ApmJSXwFUwXAva7qLK/jJJTJ2G03u3A4Oy8XGD0kn79nF6AKv9egbdD9IcIlgKZW5kc3RyZWFtCmVuZG9iagoyNyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDY4ID4+CnN0cmVhbQp4nDMyt1AwULA0ARKGFiYK5mYGCimGXEC+qYm5Qi4XSAzEygGzDIC0JZyCiFtCNEGUglgQpWYmZhBJOAMilwYAybQV5QplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDM3Ci9TdWJ0eXBlIC9Gb3JtIC9UeXBlIC9YT2JqZWN0ID4+CnN0cmVhbQp4nOMyNDBTMDY1VcjlMjc2ArNywCwjcyMgCySLYEFk0wABXwoKCmVuZHN0cmVhbQplbmRvYmoKMjkgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNjEgPj4Kc3RyZWFtCnicRZBLEsMgDEP3nEJH8EcGfJ50ukrvv60hTbOAp7FABncnBKm1BRPRBS9tS7oLPlsJzsZ46DZuNRLkBHWAVqTjaJRSfbnFaZV08Wg2cysLrRMdZg56lKMZoBA6Fd7touRypu7O+Udw9V/1R7HunM3EwGTlDoRm9SnufJsdUV3dZH/SY27Wa38V9qqwtKyl5YTbzl0zoATuqRzt/QWpczqECmVuZHN0cmVhbQplbmRvYmoKMzAgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA4MCA+PgpzdHJlYW0KeJxFjLsNwDAIRHumYAR+JmafKJWzfxsgStxwT7p7uDoSMlPeYYaHBJ4MLIZT8QaZo2A1uEZSjZ3so7BuX3WB5npTq/X3BypPdnZxPc3LGfQKZW5kc3RyZWFtCmVuZG9iagozMSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE0NyA+PgpzdHJlYW0KeJw9T7kNAzEM6z0FFzjAeixb81yQ6rJ/G8pGUggiQPGRZUfHClxiApOOORIvaT/4aRqBWAY1R/SEimFY4G6SAg+DLEpXni1eDJHaQl1I+NYQ3q1MZKI8rxE7cCcXowc+VBtZHnpAO0QVWa5Jw1jVVl1qnbACHLLOwnU9zKoE5dEnaykfUFRCvXT/n3va+wsAby/rCmVuZHN0cmVhbQplbmRvYmoKMzIgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNDkgPj4Kc3RyZWFtCnicNY9LDgMhDEP3OYUvMFJ+hHAeqq6m9982YVoJCQvbL8EWg5GMS0xg7Jhj4SVUT60+JCOPukk5EKlQNwRPaEwMM2zSJfDKdN8ynlu8nFbqgk5I5OmsNhqijGZew9FTzgqb/svcJGplRpkDMutUtxOysmAF5gW1PPcz7qhc6ISHncqw6E4xotxmRhp+/9v0/gJ7MjBjCmVuZHN0cmVhbQplbmRvYmoKMzMgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA0OSA+PgpzdHJlYW0KeJwzNrRQMFAwNDAHkkaGQJaRiUKKIRdIAMTM5YIJ5oBZBkAaojgHriaHKw0AxugNJgplbmRzdHJlYW0KZW5kb2JqCjM0IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzE3ID4+CnN0cmVhbQp4nDVSS3JDMQjbv1Nwgc6Yv32edLJq7r+thCcrsC1AQi4vWdJLftQl26XD5Fcf9yWxQj6P7ZrMUsX3FrMUzy2vR88Rty0KBFETPfgyJxUi1M/U6Dp4YZc+A68QTikWeAeTAAav4V94lE6DwDsbMt4Rk5EaECTBmkuLTUiUPUn8K+X1pJU0dH4mK3P5e3KpFGqjyQgVIFi52AekKykeJBM9iUiycr03VojekFeSx2clJhkQ3SaxTbTA49yVtISZmEIF5liA1XSzuvocTFjjsITxKmEW1YNNnjWphGa0jmNkw3j3wkyJhYbDElCbfZUJqpeP09wJI6ZHTXbtwrJbNu8hRKP5MyyUwccoJAGHTmMkCtKwgBGBOb2wir3mCzkWwIhlnZosDG1oJbt6joXA0JyzpWHG157X8/4HRVt7owplbmRzdHJlYW0KZW5kb2JqCjM1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTcgPj4Kc3RyZWFtCnicMza0UDCAwxRDLgAalALsCmVuZHN0cmVhbQplbmRvYmoKMzYgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNDggPj4Kc3RyZWFtCnicLVE5kgNBCMvnFXpCc9PvscuR9//pCsoBg4ZDIDotcVDGTxCWK97yyFW04e+ZGMF3waHfynUbFjkQFUjSGFRNqF28Hr0HdhxmAvOkNSyDGesDP2MKN3pxeEzG2e11GTUEe9drT2ZQMisXccnEBVN12MiZw0+mjAvtXM8NyLkR1mUYpJuVxoyEI00hUkih6iapM0GQBKOrUaONHMV+6csjnWFVI2oM+1xL29dzE84aNDsWqzw5pUdXnMvJxQsrB/28zcBFVBqrPBAScL/bQ/2c7OQ33tK5s8X0+F5zsrwwFVjx5rUbkE21+Dcv4vg94+v5/AOopVsWCmVuZHN0cmVhbQplbmRvYmoKMzcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyMTAgPj4Kc3RyZWFtCnicNVDLDUMxCLtnChaoFAKBZJ5WvXX/a23QO2ER/0JYyJQIeanJzinpSz46TA+2Lr+xIgutdSXsypognivvoZmysdHY4mBwGiZegBY3YOhpjRo1dOGCpi6VQoHFJfCZfHV76L5PGXhqGXJ2BBFDyWAJaroWTVi0PJ+QTgHi/37D7i3koZLzyp4b+Ruc7fA7s27hJ2p2ItFyFTLUszTHGAgTRR48eUWmcOKz1nfVNBLUZgtOlgGuTj+MDgBgIl5ZgOyuRDlL0o6ln2+8x/cPQABTtAplbmRzdHJlYW0KZW5kb2JqCjIyIDAgb2JqCjw8IC9CYXNlRm9udCAvRGVqYVZ1U2FucyAvQ2hhclByb2NzIDIzIDAgUgovRW5jb2RpbmcgPDwKL0RpZmZlcmVuY2VzIFsgMzIgL3NwYWNlIDQwIC9wYXJlbmxlZnQgL3BhcmVucmlnaHQgNDYgL3BlcmlvZCA0OCAvemVybyAvb25lIC90d28gNTIKL2ZvdXIgNTQgL3NpeCA1NiAvZWlnaHQgMTA1IC9pIDExMCAvbiAxMjQgL2JhciBdCi9UeXBlIC9FbmNvZGluZyA+PgovRmlyc3RDaGFyIDAgL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRm9udERlc2NyaXB0b3IgMjEgMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUgL05hbWUgL0RlamFWdVNhbnMKL1N1YnR5cGUgL1R5cGUzIC9UeXBlIC9Gb250IC9XaWR0aHMgMjAgMCBSID4+CmVuZG9iagoyMSAwIG9iago8PCAvQXNjZW50IDkyOSAvQ2FwSGVpZ2h0IDAgL0Rlc2NlbnQgLTIzNiAvRmxhZ3MgMzIKL0ZvbnRCQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRm9udE5hbWUgL0RlamFWdVNhbnMgL0l0YWxpY0FuZ2xlIDAKL01heFdpZHRoIDEzNDIgL1N0ZW1WIDAgL1R5cGUgL0ZvbnREZXNjcmlwdG9yIC9YSGVpZ2h0IDAgPj4KZW5kb2JqCjIwIDAgb2JqClsgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAKNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCAzMTggNDAxIDQ2MCA4MzggNjM2Cjk1MCA3ODAgMjc1IDM5MCAzOTAgNTAwIDgzOCAzMTggMzYxIDMxOCAzMzcgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNgo2MzYgNjM2IDMzNyAzMzcgODM4IDgzOCA4MzggNTMxIDEwMDAgNjg0IDY4NiA2OTggNzcwIDYzMiA1NzUgNzc1IDc1MiAyOTUKMjk1IDY1NiA1NTcgODYzIDc0OCA3ODcgNjAzIDc4NyA2OTUgNjM1IDYxMSA3MzIgNjg0IDk4OSA2ODUgNjExIDY4NSAzOTAgMzM3CjM5MCA4MzggNTAwIDUwMCA2MTMgNjM1IDU1MCA2MzUgNjE1IDM1MiA2MzUgNjM0IDI3OCAyNzggNTc5IDI3OCA5NzQgNjM0IDYxMgo2MzUgNjM1IDQxMSA1MjEgMzkyIDYzNCA1OTIgODE4IDU5MiA1OTIgNTI1IDYzNiAzMzcgNjM2IDgzOCA2MDAgNjM2IDYwMCAzMTgKMzUyIDUxOCAxMDAwIDUwMCA1MDAgNTAwIDEzNDIgNjM1IDQwMCAxMDcwIDYwMCA2ODUgNjAwIDYwMCAzMTggMzE4IDUxOCA1MTgKNTkwIDUwMCAxMDAwIDUwMCAxMDAwIDUyMSA0MDAgMTAyMyA2MDAgNTI1IDYxMSAzMTggNDAxIDYzNiA2MzYgNjM2IDYzNiAzMzcKNTAwIDUwMCAxMDAwIDQ3MSA2MTIgODM4IDM2MSAxMDAwIDUwMCA1MDAgODM4IDQwMSA0MDEgNTAwIDYzNiA2MzYgMzE4IDUwMAo0MDEgNDcxIDYxMiA5NjkgOTY5IDk2OSA1MzEgNjg0IDY4NCA2ODQgNjg0IDY4NCA2ODQgOTc0IDY5OCA2MzIgNjMyIDYzMiA2MzIKMjk1IDI5NSAyOTUgMjk1IDc3NSA3NDggNzg3IDc4NyA3ODcgNzg3IDc4NyA4MzggNzg3IDczMiA3MzIgNzMyIDczMiA2MTEgNjA1CjYzMCA2MTMgNjEzIDYxMyA2MTMgNjEzIDYxMyA5ODIgNTUwIDYxNSA2MTUgNjE1IDYxNSAyNzggMjc4IDI3OCAyNzggNjEyIDYzNAo2MTIgNjEyIDYxMiA2MTIgNjEyIDgzOCA2MTIgNjM0IDYzNCA2MzQgNjM0IDU5MiA2MzUgNTkyIF0KZW5kb2JqCjIzIDAgb2JqCjw8IC9iYXIgMjQgMCBSIC9laWdodCAyNSAwIFIgL2ZvdXIgMjYgMCBSIC9pIDI3IDAgUiAvbiAyOSAwIFIgL29uZSAzMCAwIFIKL3BhcmVubGVmdCAzMSAwIFIgL3BhcmVucmlnaHQgMzIgMCBSIC9wZXJpb2QgMzMgMCBSIC9zaXggMzQgMCBSCi9zcGFjZSAzNSAwIFIgL3R3byAzNiAwIFIgL3plcm8gMzcgMCBSID4+CmVuZG9iagozIDAgb2JqCjw8IC9GMSAyMiAwIFIgL0YyIDE0IDAgUiA+PgplbmRvYmoKNCAwIG9iago8PCAvQTEgPDwgL0NBIDAgL1R5cGUgL0V4dEdTdGF0ZSAvY2EgMSA+PgovQTIgPDwgL0NBIDEgL1R5cGUgL0V4dEdTdGF0ZSAvY2EgMSA+PiA+PgplbmRvYmoKNSAwIG9iago8PCA+PgplbmRvYmoKNiAwIG9iago8PCA+PgplbmRvYmoKNyAwIG9iago8PCAvRGVqYVZ1U2Fucy1PYmxpcXVlLW9tZWdhIDE4IDAgUiAvRGVqYVZ1U2Fucy1taW51cyAyOCAwIFIgPj4KZW5kb2JqCjIgMCBvYmoKPDwgL0NvdW50IDEgL0tpZHMgWyAxMCAwIFIgXSAvVHlwZSAvUGFnZXMgPj4KZW5kb2JqCjM4IDAgb2JqCjw8IC9DcmVhdGlvbkRhdGUgKEQ6MjAxOTA1MjkwOTM3MDYrMDInMDAnKQovQ3JlYXRvciAobWF0cGxvdGxpYiAzLjAuMywgaHR0cDovL21hdHBsb3RsaWIub3JnKQovUHJvZHVjZXIgKG1hdHBsb3RsaWIgcGRmIGJhY2tlbmQgMy4wLjMpID4+CmVuZG9iagp4cmVmCjAgMzkKMDAwMDAwMDAwMCA2NTUzNSBmIAowMDAwMDAwMDE2IDAwMDAwIG4gCjAwMDAwMTAzNzUgMDAwMDAgbiAKMDAwMDAxMDExMiAwMDAwMCBuIAowMDAwMDEwMTU1IDAwMDAwIG4gCjAwMDAwMTAyNTQgMDAwMDAgbiAKMDAwMDAxMDI3NSAwMDAwMCBuIAowMDAwMDEwMjk2IDAwMDAwIG4gCjAwMDAwMDAwNjUgMDAwMDAgbiAKMDAwMDAwMDM5OCAwMDAwMCBuIAowMDAwMDAwMjA4IDAwMDAwIG4gCjAwMDAwMDIzOTggMDAwMDAgbiAKMDAwMDAwNDA1NCAwMDAwMCBuIAowMDAwMDAzODQ2IDAwMDAwIG4gCjAwMDAwMDM1MTcgMDAwMDAgbiAKMDAwMDAwNTEwNyAwMDAwMCBuIAowMDAwMDAyNDE5IDAwMDAwIG4gCjAwMDAwMDI1NzkgMDAwMDAgbiAKMDAwMDAwMjgwNCAwMDAwMCBuIAowMDAwMDAzMTE5IDAwMDAwIG4gCjAwMDAwMDg4NjMgMDAwMDAgbiAKMDAwMDAwODY2MyAwMDAwMCBuIAowMDAwMDA4MjU0IDAwMDAwIG4gCjAwMDAwMDk5MTYgMDAwMDAgbiAKMDAwMDAwNTE1OSAwMDAwMCBuIAowMDAwMDA1Mjg1IDAwMDAwIG4gCjAwMDAwMDU3NTAgMDAwMDAgbiAKMDAwMDAwNTkxMiAwMDAwMCBuIAowMDAwMDA2MDUyIDAwMDAwIG4gCjAwMDAwMDYyMjIgMDAwMDAgbiAKMDAwMDAwNjQ1NiAwMDAwMCBuIAowMDAwMDA2NjA4IDAwMDAwIG4gCjAwMDAwMDY4MjggMDAwMDAgbiAKMDAwMDAwNzA1MCAwMDAwMCBuIAowMDAwMDA3MTcxIDAwMDAwIG4gCjAwMDAwMDc1NjEgMDAwMDAgbiAKMDAwMDAwNzY1MCAwMDAwMCBuIAowMDAwMDA3OTcxIDAwMDAwIG4gCjAwMDAwMTA0MzUgMDAwMDAgbiAKdHJhaWxlcgo8PCAvSW5mbyAzOCAwIFIgL1Jvb3QgMSAwIFIgL1NpemUgMzkgPj4Kc3RhcnR4cmVmCjEwNTg5CiUlRU9GCg==\n",
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (https://matplotlib.org/) -->\n",
       "<svg height=\"314.36825pt\" version=\"1.1\" viewBox=\"0 0 412.023125 314.36825\" width=\"412.023125pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 314.36825 \n",
       "L 412.023125 314.36825 \n",
       "L 412.023125 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 44.203125 276.812 \n",
       "L 401.323125 276.812 \n",
       "L 401.323125 10.7 \n",
       "L 44.203125 10.7 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 57.156513 276.812 \n",
       "L 57.156513 10.7 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_2\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" id=\"mb7dee615cb\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"57.156513\" xlink:href=\"#mb7dee615cb\" y=\"276.812\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0 -->\n",
       "      <defs>\n",
       "       <path d=\"M 31.78125 66.40625 \n",
       "Q 24.171875 66.40625 20.328125 58.90625 \n",
       "Q 16.5 51.421875 16.5 36.375 \n",
       "Q 16.5 21.390625 20.328125 13.890625 \n",
       "Q 24.171875 6.390625 31.78125 6.390625 \n",
       "Q 39.453125 6.390625 43.28125 13.890625 \n",
       "Q 47.125 21.390625 47.125 36.375 \n",
       "Q 47.125 51.421875 43.28125 58.90625 \n",
       "Q 39.453125 66.40625 31.78125 66.40625 \n",
       "z\n",
       "M 31.78125 74.21875 \n",
       "Q 44.046875 74.21875 50.515625 64.515625 \n",
       "Q 56.984375 54.828125 56.984375 36.375 \n",
       "Q 56.984375 17.96875 50.515625 8.265625 \n",
       "Q 44.046875 -1.421875 31.78125 -1.421875 \n",
       "Q 19.53125 -1.421875 13.0625 8.265625 \n",
       "Q 6.59375 17.96875 6.59375 36.375 \n",
       "Q 6.59375 54.828125 13.0625 64.515625 \n",
       "Q 19.53125 74.21875 31.78125 74.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-48\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(53.975263 291.410438)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 122.74329 276.812 \n",
       "L 122.74329 10.7 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"122.74329\" xlink:href=\"#mb7dee615cb\" y=\"276.812\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 2 -->\n",
       "      <defs>\n",
       "       <path d=\"M 19.1875 8.296875 \n",
       "L 53.609375 8.296875 \n",
       "L 53.609375 0 \n",
       "L 7.328125 0 \n",
       "L 7.328125 8.296875 \n",
       "Q 12.9375 14.109375 22.625 23.890625 \n",
       "Q 32.328125 33.6875 34.8125 36.53125 \n",
       "Q 39.546875 41.84375 41.421875 45.53125 \n",
       "Q 43.3125 49.21875 43.3125 52.78125 \n",
       "Q 43.3125 58.59375 39.234375 62.25 \n",
       "Q 35.15625 65.921875 28.609375 65.921875 \n",
       "Q 23.96875 65.921875 18.8125 64.3125 \n",
       "Q 13.671875 62.703125 7.8125 59.421875 \n",
       "L 7.8125 69.390625 \n",
       "Q 13.765625 71.78125 18.9375 73 \n",
       "Q 24.125 74.21875 28.421875 74.21875 \n",
       "Q 39.75 74.21875 46.484375 68.546875 \n",
       "Q 53.21875 62.890625 53.21875 53.421875 \n",
       "Q 53.21875 48.921875 51.53125 44.890625 \n",
       "Q 49.859375 40.875 45.40625 35.40625 \n",
       "Q 44.1875 33.984375 37.640625 27.21875 \n",
       "Q 31.109375 20.453125 19.1875 8.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-50\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(119.56204 291.410438)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 188.330067 276.812 \n",
       "L 188.330067 10.7 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"188.330067\" xlink:href=\"#mb7dee615cb\" y=\"276.812\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 4 -->\n",
       "      <defs>\n",
       "       <path d=\"M 37.796875 64.3125 \n",
       "L 12.890625 25.390625 \n",
       "L 37.796875 25.390625 \n",
       "z\n",
       "M 35.203125 72.90625 \n",
       "L 47.609375 72.90625 \n",
       "L 47.609375 25.390625 \n",
       "L 58.015625 25.390625 \n",
       "L 58.015625 17.1875 \n",
       "L 47.609375 17.1875 \n",
       "L 47.609375 0 \n",
       "L 37.796875 0 \n",
       "L 37.796875 17.1875 \n",
       "L 4.890625 17.1875 \n",
       "L 4.890625 26.703125 \n",
       "z\n",
       "\" id=\"DejaVuSans-52\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(185.148817 291.410438)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-52\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 253.916844 276.812 \n",
       "L 253.916844 10.7 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"253.916844\" xlink:href=\"#mb7dee615cb\" y=\"276.812\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 6 -->\n",
       "      <defs>\n",
       "       <path d=\"M 33.015625 40.375 \n",
       "Q 26.375 40.375 22.484375 35.828125 \n",
       "Q 18.609375 31.296875 18.609375 23.390625 \n",
       "Q 18.609375 15.53125 22.484375 10.953125 \n",
       "Q 26.375 6.390625 33.015625 6.390625 \n",
       "Q 39.65625 6.390625 43.53125 10.953125 \n",
       "Q 47.40625 15.53125 47.40625 23.390625 \n",
       "Q 47.40625 31.296875 43.53125 35.828125 \n",
       "Q 39.65625 40.375 33.015625 40.375 \n",
       "z\n",
       "M 52.59375 71.296875 \n",
       "L 52.59375 62.3125 \n",
       "Q 48.875 64.0625 45.09375 64.984375 \n",
       "Q 41.3125 65.921875 37.59375 65.921875 \n",
       "Q 27.828125 65.921875 22.671875 59.328125 \n",
       "Q 17.53125 52.734375 16.796875 39.40625 \n",
       "Q 19.671875 43.65625 24.015625 45.921875 \n",
       "Q 28.375 48.1875 33.59375 48.1875 \n",
       "Q 44.578125 48.1875 50.953125 41.515625 \n",
       "Q 57.328125 34.859375 57.328125 23.390625 \n",
       "Q 57.328125 12.15625 50.6875 5.359375 \n",
       "Q 44.046875 -1.421875 33.015625 -1.421875 \n",
       "Q 20.359375 -1.421875 13.671875 8.265625 \n",
       "Q 6.984375 17.96875 6.984375 36.375 \n",
       "Q 6.984375 53.65625 15.1875 63.9375 \n",
       "Q 23.390625 74.21875 37.203125 74.21875 \n",
       "Q 40.921875 74.21875 44.703125 73.484375 \n",
       "Q 48.484375 72.75 52.59375 71.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-54\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(250.735594 291.410438)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-54\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 319.503621 276.812 \n",
       "L 319.503621 10.7 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"319.503621\" xlink:href=\"#mb7dee615cb\" y=\"276.812\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 8 -->\n",
       "      <defs>\n",
       "       <path d=\"M 31.78125 34.625 \n",
       "Q 24.75 34.625 20.71875 30.859375 \n",
       "Q 16.703125 27.09375 16.703125 20.515625 \n",
       "Q 16.703125 13.921875 20.71875 10.15625 \n",
       "Q 24.75 6.390625 31.78125 6.390625 \n",
       "Q 38.8125 6.390625 42.859375 10.171875 \n",
       "Q 46.921875 13.96875 46.921875 20.515625 \n",
       "Q 46.921875 27.09375 42.890625 30.859375 \n",
       "Q 38.875 34.625 31.78125 34.625 \n",
       "z\n",
       "M 21.921875 38.8125 \n",
       "Q 15.578125 40.375 12.03125 44.71875 \n",
       "Q 8.5 49.078125 8.5 55.328125 \n",
       "Q 8.5 64.0625 14.71875 69.140625 \n",
       "Q 20.953125 74.21875 31.78125 74.21875 \n",
       "Q 42.671875 74.21875 48.875 69.140625 \n",
       "Q 55.078125 64.0625 55.078125 55.328125 \n",
       "Q 55.078125 49.078125 51.53125 44.71875 \n",
       "Q 48 40.375 41.703125 38.8125 \n",
       "Q 48.828125 37.15625 52.796875 32.3125 \n",
       "Q 56.78125 27.484375 56.78125 20.515625 \n",
       "Q 56.78125 9.90625 50.3125 4.234375 \n",
       "Q 43.84375 -1.421875 31.78125 -1.421875 \n",
       "Q 19.734375 -1.421875 13.25 4.234375 \n",
       "Q 6.78125 9.90625 6.78125 20.515625 \n",
       "Q 6.78125 27.484375 10.78125 32.3125 \n",
       "Q 14.796875 37.15625 21.921875 38.8125 \n",
       "z\n",
       "M 18.3125 54.390625 \n",
       "Q 18.3125 48.734375 21.84375 45.5625 \n",
       "Q 25.390625 42.390625 31.78125 42.390625 \n",
       "Q 38.140625 42.390625 41.71875 45.5625 \n",
       "Q 45.3125 48.734375 45.3125 54.390625 \n",
       "Q 45.3125 60.0625 41.71875 63.234375 \n",
       "Q 38.140625 66.40625 31.78125 66.40625 \n",
       "Q 25.390625 66.40625 21.84375 63.234375 \n",
       "Q 18.3125 60.0625 18.3125 54.390625 \n",
       "z\n",
       "\" id=\"DejaVuSans-56\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(316.322371 291.410438)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-56\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 385.090398 276.812 \n",
       "L 385.090398 10.7 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"385.090398\" xlink:href=\"#mb7dee615cb\" y=\"276.812\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 10 -->\n",
       "      <defs>\n",
       "       <path d=\"M 12.40625 8.296875 \n",
       "L 28.515625 8.296875 \n",
       "L 28.515625 63.921875 \n",
       "L 10.984375 60.40625 \n",
       "L 10.984375 69.390625 \n",
       "L 28.421875 72.90625 \n",
       "L 38.28125 72.90625 \n",
       "L 38.28125 8.296875 \n",
       "L 54.390625 8.296875 \n",
       "L 54.390625 0 \n",
       "L 12.40625 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-49\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(378.727898 291.410438)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- $\\omega$ in $s^{-1}$ -->\n",
       "     <defs>\n",
       "      <path d=\"M 21.390625 -1.421875 \n",
       "Q 1.125 -1.421875 6.78125 27.59375 \n",
       "Q 8.984375 39.0625 19.625 54.6875 \n",
       "L 29.296875 54.6875 \n",
       "Q 19.234375 39.0625 17 27.34375 \n",
       "Q 12.796875 6.203125 23.734375 6.203125 \n",
       "Q 33.84375 6.203125 38.71875 32.46875 \n",
       "L 47.015625 32.46875 \n",
       "Q 41.703125 6.0625 51.8125 6.203125 \n",
       "Q 62.703125 6.296875 66.796875 27.34375 \n",
       "Q 69.046875 39.0625 65.140625 54.6875 \n",
       "L 74.8125 54.6875 \n",
       "Q 79.296875 39.0625 77.09375 27.59375 \n",
       "Q 71.53125 -1.46875 51.21875 -1.421875 \n",
       "Q 37.9375 -1.375 39.109375 13.1875 \n",
       "Q 34.28125 -1.421875 21.390625 -1.421875 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-969\"/>\n",
       "      <path id=\"DejaVuSans-32\"/>\n",
       "      <path d=\"M 9.421875 54.6875 \n",
       "L 18.40625 54.6875 \n",
       "L 18.40625 0 \n",
       "L 9.421875 0 \n",
       "z\n",
       "M 9.421875 75.984375 \n",
       "L 18.40625 75.984375 \n",
       "L 18.40625 64.59375 \n",
       "L 9.421875 64.59375 \n",
       "z\n",
       "\" id=\"DejaVuSans-105\"/>\n",
       "      <path d=\"M 54.890625 33.015625 \n",
       "L 54.890625 0 \n",
       "L 45.90625 0 \n",
       "L 45.90625 32.71875 \n",
       "Q 45.90625 40.484375 42.875 44.328125 \n",
       "Q 39.84375 48.1875 33.796875 48.1875 \n",
       "Q 26.515625 48.1875 22.3125 43.546875 \n",
       "Q 18.109375 38.921875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.34375 51.125 25.703125 53.5625 \n",
       "Q 30.078125 56 35.796875 56 \n",
       "Q 45.21875 56 50.046875 50.171875 \n",
       "Q 54.890625 44.34375 54.890625 33.015625 \n",
       "z\n",
       "\" id=\"DejaVuSans-110\"/>\n",
       "      <path d=\"M 50 53.078125 \n",
       "L 48.296875 44.578125 \n",
       "Q 44.734375 46.53125 40.765625 47.5 \n",
       "Q 36.8125 48.484375 32.625 48.484375 \n",
       "Q 25.53125 48.484375 21.453125 46.0625 \n",
       "Q 17.390625 43.65625 17.390625 39.5 \n",
       "Q 17.390625 34.671875 26.859375 32.078125 \n",
       "Q 27.59375 31.890625 27.9375 31.78125 \n",
       "L 30.8125 30.90625 \n",
       "Q 39.796875 28.421875 42.796875 25.6875 \n",
       "Q 45.796875 22.953125 45.796875 18.21875 \n",
       "Q 45.796875 9.515625 38.890625 4.046875 \n",
       "Q 31.984375 -1.421875 20.796875 -1.421875 \n",
       "Q 16.453125 -1.421875 11.671875 -0.578125 \n",
       "Q 6.890625 0.25 1.125 2 \n",
       "L 2.875 11.28125 \n",
       "Q 7.8125 8.734375 12.59375 7.421875 \n",
       "Q 17.390625 6.109375 21.78125 6.109375 \n",
       "Q 28.375 6.109375 32.5 8.9375 \n",
       "Q 36.625 11.765625 36.625 16.109375 \n",
       "Q 36.625 20.796875 25.78125 23.6875 \n",
       "L 24.859375 23.921875 \n",
       "L 21.78125 24.703125 \n",
       "Q 14.9375 26.515625 11.765625 29.46875 \n",
       "Q 8.59375 32.421875 8.59375 37.015625 \n",
       "Q 8.59375 45.75 15.15625 50.875 \n",
       "Q 21.734375 56 33.015625 56 \n",
       "Q 37.453125 56 41.671875 55.265625 \n",
       "Q 45.90625 54.546875 50 53.078125 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-115\"/>\n",
       "      <path d=\"M 10.59375 35.5 \n",
       "L 73.1875 35.5 \n",
       "L 73.1875 27.203125 \n",
       "L 10.59375 27.203125 \n",
       "z\n",
       "\" id=\"DejaVuSans-8722\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(202.663125 305.088562)scale(0.1 -0.1)\">\n",
       "      <use transform=\"translate(0 0.684375)\" xlink:href=\"#DejaVuSans-Oblique-969\"/>\n",
       "      <use transform=\"translate(83.740234 0.684375)\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use transform=\"translate(115.527344 0.684375)\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use transform=\"translate(143.310547 0.684375)\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use transform=\"translate(206.689453 0.684375)\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use transform=\"translate(238.476562 0.684375)\" xlink:href=\"#DejaVuSans-Oblique-115\"/>\n",
       "      <use transform=\"translate(295.226068 38.965625)scale(0.7)\" xlink:href=\"#DejaVuSans-8722\"/>\n",
       "      <use transform=\"translate(353.878411 38.965625)scale(0.7)\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 275.111946 \n",
       "L 401.323125 275.111946 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_14\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" id=\"mce3e61c8e3\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"275.111946\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 0.0 -->\n",
       "      <defs>\n",
       "       <path d=\"M 10.6875 12.40625 \n",
       "L 21 12.40625 \n",
       "L 21 0 \n",
       "L 10.6875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-46\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(21.3 278.911165)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 234.741394 \n",
       "L 401.323125 234.741394 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"234.741394\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(21.3 238.540613)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 194.370843 \n",
       "L 401.323125 194.370843 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"194.370843\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(21.3 198.170061)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-52\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_19\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 154.000291 \n",
       "L 401.323125 154.000291 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_20\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"154.000291\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(21.3 157.79951)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-54\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_21\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 113.629739 \n",
       "L 401.323125 113.629739 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_22\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"113.629739\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(21.3 117.428958)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-56\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_23\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 73.259188 \n",
       "L 401.323125 73.259188 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_24\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"73.259188\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(21.3 77.058406)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_25\">\n",
       "      <path clip-path=\"url(#pfc21d794ba)\" d=\"M 44.203125 32.888636 \n",
       "L 401.323125 32.888636 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_26\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"44.203125\" xlink:href=\"#mce3e61c8e3\" y=\"32.888636\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_14\">\n",
       "      <!-- 1.2 -->\n",
       "      <g transform=\"translate(21.3 36.687855)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n",
       "       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_15\">\n",
       "     <!-- $|H(j \\omega)|$ -->\n",
       "     <defs>\n",
       "      <path d=\"M 21 76.421875 \n",
       "L 21 -23.578125 \n",
       "L 12.703125 -23.578125 \n",
       "L 12.703125 76.421875 \n",
       "z\n",
       "\" id=\"DejaVuSans-124\"/>\n",
       "      <path d=\"M 16.890625 72.90625 \n",
       "L 26.8125 72.90625 \n",
       "L 21 43.015625 \n",
       "L 56.78125 43.015625 \n",
       "L 62.59375 72.90625 \n",
       "L 72.515625 72.90625 \n",
       "L 58.296875 0 \n",
       "L 48.390625 0 \n",
       "L 55.171875 34.71875 \n",
       "L 19.390625 34.71875 \n",
       "L 12.59375 0 \n",
       "L 2.6875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-72\"/>\n",
       "      <path d=\"M 31 75.875 \n",
       "Q 24.46875 64.65625 21.28125 53.65625 \n",
       "Q 18.109375 42.671875 18.109375 31.390625 \n",
       "Q 18.109375 20.125 21.3125 9.0625 \n",
       "Q 24.515625 -2 31 -13.1875 \n",
       "L 23.1875 -13.1875 \n",
       "Q 15.875 -1.703125 12.234375 9.375 \n",
       "Q 8.59375 20.453125 8.59375 31.390625 \n",
       "Q 8.59375 42.28125 12.203125 53.3125 \n",
       "Q 15.828125 64.359375 23.1875 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-40\"/>\n",
       "      <path d=\"M 14.5 54.6875 \n",
       "L 23.484375 54.6875 \n",
       "L 12.703125 -0.984375 \n",
       "L 12.640625 -1.21875 \n",
       "Q 10.84375 -10.546875 8.5 -14.015625 \n",
       "Q 6.296875 -17.28125 2.0625 -19.03125 \n",
       "Q -2.15625 -20.796875 -7.90625 -20.796875 \n",
       "L -11.28125 -20.796875 \n",
       "L -9.8125 -13.1875 \n",
       "L -7.515625 -13.1875 \n",
       "Q -2.25 -13.1875 -0.015625 -10.984375 \n",
       "Q 2.203125 -8.796875 3.71875 -0.984375 \n",
       "z\n",
       "M 18.703125 75.984375 \n",
       "L 27.6875 75.984375 \n",
       "L 25.484375 64.59375 \n",
       "L 16.5 64.59375 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-106\"/>\n",
       "      <path d=\"M 8.015625 75.875 \n",
       "L 15.828125 75.875 \n",
       "Q 23.140625 64.359375 26.78125 53.3125 \n",
       "Q 30.421875 42.28125 30.421875 31.390625 \n",
       "Q 30.421875 20.453125 26.78125 9.375 \n",
       "Q 23.140625 -1.703125 15.828125 -13.1875 \n",
       "L 8.015625 -13.1875 \n",
       "Q 14.5 -2 17.703125 9.0625 \n",
       "Q 20.90625 20.125 20.90625 31.390625 \n",
       "Q 20.90625 42.671875 17.703125 53.65625 \n",
       "Q 14.5 64.65625 8.015625 75.875 \n",
       "z\n",
       "\" id=\"DejaVuSans-41\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(14.9 160.406)rotate(-90)scale(0.1 -0.1)\">\n",
       "      <use transform=\"translate(0 0.578125)\" xlink:href=\"#DejaVuSans-124\"/>\n",
       "      <use transform=\"translate(33.691406 0.578125)\" xlink:href=\"#DejaVuSans-Oblique-72\"/>\n",
       "      <use transform=\"translate(108.886719 0.578125)\" xlink:href=\"#DejaVuSans-40\"/>\n",
       "      <use transform=\"translate(147.900391 0.578125)\" xlink:href=\"#DejaVuSans-Oblique-106\"/>\n",
       "      <use transform=\"translate(175.683594 0.578125)\" xlink:href=\"#DejaVuSans-Oblique-969\"/>\n",
       "      <use transform=\"translate(259.423828 0.578125)\" xlink:href=\"#DejaVuSans-41\"/>\n",
       "      <use transform=\"translate(298.4375 0.578125)\" xlink:href=\"#DejaVuSans-124\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_27\">\n",
       "    <path clip-path=\"url(#pfc21d794ba)\" d=\"M 60.435852 73.016933 \n",
       "L 62.66724 72.574935 \n",
       "L 64.919409 71.90093 \n",
       "L 67.198853 70.985335 \n",
       "L 69.522932 69.809843 \n",
       "L 71.894955 68.358348 \n",
       "L 74.329989 66.604716 \n",
       "L 76.847349 64.513568 \n",
       "L 79.4547 62.054726 \n",
       "L 82.198835 59.155304 \n",
       "L 85.112792 55.746461 \n",
       "L 88.308379 51.656949 \n",
       "L 92.061623 46.476189 \n",
       "L 103.661708 30.170644 \n",
       "L 105.898531 27.586963 \n",
       "L 107.751225 25.770262 \n",
       "L 109.360838 24.489592 \n",
       "L 110.798847 23.621022 \n",
       "L 112.09905 23.089143 \n",
       "L 113.301327 22.833303 \n",
       "L 114.424314 22.815424 \n",
       "L 115.515982 23.01565 \n",
       "L 116.601075 23.437949 \n",
       "L 117.706343 24.106044 \n",
       "L 118.860574 25.06776 \n",
       "L 120.065775 26.365557 \n",
       "L 121.353622 28.086714 \n",
       "L 122.728205 30.303049 \n",
       "L 124.224824 33.151844 \n",
       "L 125.85045 36.738859 \n",
       "L 127.645218 41.253152 \n",
       "L 129.687029 47.019791 \n",
       "L 132.028981 54.327879 \n",
       "L 134.910701 64.086193 \n",
       "L 139.139944 79.285858 \n",
       "L 147.091797 107.922983 \n",
       "L 150.941199 120.878041 \n",
       "L 154.37156 131.664278 \n",
       "L 157.649309 141.244679 \n",
       "L 160.798629 149.772883 \n",
       "L 163.899057 157.536582 \n",
       "L 166.991016 164.683587 \n",
       "L 170.068484 171.24273 \n",
       "L 173.178733 177.349988 \n",
       "L 176.319761 183.027399 \n",
       "L 179.489469 188.29946 \n",
       "L 182.685664 193.191772 \n",
       "L 185.965365 197.810212 \n",
       "L 189.330756 202.166507 \n",
       "L 192.784074 206.272927 \n",
       "L 196.327616 210.141974 \n",
       "L 199.963742 213.786143 \n",
       "L 203.762373 217.277162 \n",
       "L 207.662048 220.56059 \n",
       "L 211.73663 223.70113 \n",
       "L 215.99466 226.700944 \n",
       "L 220.445168 229.562682 \n",
       "L 225.097705 232.28939 \n",
       "L 229.962375 234.884448 \n",
       "L 235.13182 237.389339 \n",
       "L 240.624828 239.800358 \n",
       "L 246.461664 242.114707 \n",
       "L 252.664197 244.330435 \n",
       "L 259.256028 246.446373 \n",
       "L 266.358966 248.488259 \n",
       "L 273.911108 250.426114 \n",
       "L 282.046393 252.283339 \n",
       "L 290.917263 254.076647 \n",
       "L 300.474006 255.779461 \n",
       "L 310.88846 257.407477 \n",
       "L 322.358683 258.971123 \n",
       "L 334.858555 260.448232 \n",
       "L 348.752265 261.86199 \n",
       "L 364.188366 263.203649 \n",
       "L 381.336196 264.466592 \n",
       "L 385.090398 264.716 \n",
       "L 385.090398 264.716 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-linecap:square;stroke-width:1.5;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 44.203125 276.812 \n",
       "L 44.203125 10.7 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 401.323125 276.812 \n",
       "L 401.323125 10.7 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 44.203125 276.812 \n",
       "L 401.323125 276.812 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 44.203125 10.7 \n",
       "L 401.323125 10.7 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pfc21d794ba\">\n",
       "   <rect height=\"266.112\" width=\"357.12\" x=\"44.203125\" y=\"10.7\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.plot(w, abs(H))\n",
    "plt.xlabel('$\\omega$ in $s^{-1}$')\n",
    "plt.ylabel('$|H(j \\omega)|$')\n",
    "plt.grid();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Determination from Internal Structure\n",
    "\n",
    "The transfer function $H(s)$ can be determined from the internal structure of an LTI system, if known. Either by derivation of the ODE characterizing the system, as shown above, or by direct analysis of the system in the Laplace/Fourier domain. For passive electrical networks this can be conducted by representing the signals and elements of the network by their complex impedances $Z(s)$ and applying Kirchhoff's circuit laws. This was illustrated for instance by the previous [Laplace domain analysis of a 2nd order low-pass filter](../laplace_transform/network_analysis.ipynb#Example:-Second-Order-Low-Pass-Filter). Similar procedures can be applied to other types of LTI systems, e.g. mechanical systems.\n",
    "\n",
    "The transfer function is then given as the quotient between the output signal $Y(s)$ and the input signal $X(s)$ in the Laplace/Fourier domain."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}
